In

prosthaphaeresis
Prosthaphaeresis (from the Greek ''προσθαφαίρεσις'') was an algorithm used in the late 16th century and early 17th century for approximate multiplication and Division (mathematics), division using formulas from trigonometry. For the ...

, which relies on

natural logarithm
The natural logarithm of a number is its logarithm to the base (exponentiation), base of the mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approximately equal to . The natur ...

of can be defined as the natural logarithm
The natural logarithm of a number is its logarithm to the base (exponentiation), base of the mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approximately equal to . The natur ...

: as tends to

natural logarithm
The natural logarithm of a number is its logarithm to the base (exponentiation), base of the mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approximately equal to . The natur ...

at . The Taylor series of provides a particularly useful approximation to when is small, , since then
:$\backslash ln\; (1+z)\; =\; z\; -\; \backslash frac\; +\backslash frac\backslash cdots\; \backslash approx\; z.$
For example, with the first-order approximation gives , which is less than 5% off the correct value 0.0953.
Although the sequence for $\backslash ln(1+z)$ only converges for $,\; z,\; <1$, a neat trick can fix this.
:$\backslash ln(1+z)\; =\; -\backslash ln\backslash left(\backslash frac\backslash right)\; =\; -\backslash ln\backslash left(1-\backslash frac\backslash right)$
As $\backslash left,\; \backslash frac\backslash <1$ for all $,\; z,\; \backslash ge0$, the sequence converges for the same range of .

natural logarithm
The natural logarithm of a number is its logarithm to the base (exponentiation), base of the mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approximately equal to . The natur ...

. Sasaki and Kanada showed in 1982 that it was particularly fast for precisions between 400 and 1000 decimal places, while Taylor series methods were typically faster when less precision was needed. In their work is approximated to a precision of (or precise bits) by the following formula (due to

^{−7} mol·L^{−1}, hence a pH of 7. Vinegar typically has a pH of about 3. The difference of 4 corresponds to a ratio of 10^{4} of the activity, that is, vinegar's hydronium ion activity is about .
Semilog (log–linear) graphs use the logarithmic scale concept for visualization: one axis, typically the vertical one, is scaled logarithmically. For example, the chart at the right compresses the steep increase from 1 million to 1 trillion to the same space (on the vertical axis) as the increase from 1 to 1 million. In such graphs, exponential function
The exponential function is a mathematical Function (mathematics), function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponentiation, exponent). Unless otherwise specified, the term generally refers to the positiv ...

s of the form appear as straight lines with

natural logarithm
The natural logarithm of a number is its logarithm to the base (exponentiation), base of the mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approximately equal to . The natur ...

.
The logarithm of ''n''

natural logarithm
The natural logarithm of a number is its logarithm to the base (exponentiation), base of the mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approximately equal to . The natur ...

by . Moreover, equals the

Khan Academy: Logarithms, free online micro lectures

* * * * {{Authority control Elementary special functions Scottish inventions Additive functions

mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the logarithm is the inverse function
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

to exponentiation
Exponentiation is a mathematics, mathematical operation (mathematics), operation, written as , involving two numbers, the ''Base (exponentiation), base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". W ...

. That means the logarithm of a number to the base is the exponent
Exponentiation is a mathematics, mathematical operation (mathematics), operation, written as , involving two numbers, the ''Base (exponentiation), base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". W ...

to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of is , or . The logarithm of to ''base'' is denoted as , or without parentheses, , or even without the explicit base, , when no confusion is possible, or when the base does not matter such as in big O notation
Big ''O'' notation is a mathematical notation that describes the asymptotic analysis, limiting behavior of a function (mathematics), function when the Argument of a function, argument tends towards a particular value or infinity. Big O is a memb ...

.
The logarithm base is called the decimal or common logarithm
In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs (mathematician), Henry Briggs, an English ...

and is commonly used in science and engineering. The natural logarithm
The natural logarithm of a number is its logarithm to the base (exponentiation), base of the mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approximately equal to . The natur ...

has the number as its base; its use is widespread in mathematics and physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...

, because of its very simple derivative
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...

. The binary logarithm
In mathematics, the binary logarithm () is the exponentiation, power to which the number must be exponentiation, raised to obtain the value . That is, for any real number ,
:x=\log_2 n \quad\Longleftrightarrow\quad 2^x=n.
For example, the ...

uses base and is frequently used in computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...

.
Logarithms were introduced by John Napier
John Napier of Merchiston (; 1 February 1550 – 4 April 1617), nicknamed Marvellous Merchiston, was a Scottish landowner known as a mathematician, physicist, and astronomer. He was the 8th Laird of Merchiston. His Latinisation of names, L ...

in 1614 as a means of simplifying calculations. They were rapidly adopted by navigators, scientists, engineers, surveyors and others to perform high-accuracy computations more easily. Using logarithm tables, tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition. This is possible because the logarithm of a product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Pr ...

is the sum of the logarithms of the factors:
:$\backslash log\_b(xy)\; =\; \backslash log\_b\; x\; +\; \backslash log\_b\; y,$
provided that , and are all positive and . The slide rule
The slide rule is a mechanical analog computer which is used primarily for multiplication and division (mathematics), division, and for functions such as exponents, Nth root, roots, logarithms, and trigonometry. It is not typically designed for ...

, also based on logarithms, allows quick calculations without tables, but at lower precision. The present-day notion of logarithms comes from Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

, who connected them to the exponential function
The exponential function is a mathematical Function (mathematics), function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponentiation, exponent). Unless otherwise specified, the term generally refers to the positiv ...

in the 18th century, and who also introduced the letter as the base of natural logarithms.
Logarithmic scale
A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers. Such a ...

s reduce wide-ranging quantities to smaller scopes. For example, the decibel
The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a Power, root-power, and field quantities, power or root-power quantity on a logarithmic scale. Two signals whose ...

(dB) is a unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* Unit (album), ...

used to express ratio as logarithms, mostly for signal power and amplitude (of which sound pressure
Sound pressure or acoustic pressure is the local pressure deviation from the ambient (average or equilibrium) atmospheric pressure, caused by a sound wave. In air, sound pressure can be measured using a microphone, and in water with a hydrophone ...

is a common example). In chemistry, pH is a logarithmic measure for the acid
An acid is a molecule or ion capable of either donating a proton (i.e. hydrogen ion, H+), known as a Brønsted–Lowry acid, or forming a covalent bond with an electron pair, known as a Lewis acid
A Lewis acid (named for the American p ...

ity of an aqueous solution
An aqueous solution is a Solution (chemistry), solution in which the solvent is water. It is mostly shown in chemical equations by appending (aq) to the relevant chemical formula. For example, a solution of table salt, or sodium chloride (NaCl), ...

. Logarithms are commonplace in scientific formula
In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the terminology, term ''formula'' in science refers to the Commensurability (philosophy o ...

e, and in measurements of the complexity of algorithms and of geometric objects called fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illus ...

s. They help to describe frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in Hertz (unit), hertz (H ...

ratios of musical intervals, appear in formulas counting prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only wa ...

s or approximating factorial
In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times ...

s, inform some models in psychophysics
Psychophysics quantitatively investigates the relationship between physical stimulus (physiology), stimuli and the sensation (psychology), sensations and perceptions they produce. Psychophysics has been described as "the scientific study of the ...

, and can aid in forensic accounting.
The concept of logarithm as the inverse of exponentiation extends to other mathematical structures as well. However, in general settings, the logarithm tends to be a multi-valued function. For example, the complex logarithm
In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related:
* A complex logarithm of a nonzero complex number z, defined to be ...

is the multi-valued inverse of the complex exponential function. Similarly, the discrete logarithm
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...

is the multi-valued inverse of the exponential function in finite groups; it has uses in public-key cryptography
Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. Key pairs are generated with cryptographic alg ...

.
Motivation

Addition
Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...

, multiplication
Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...

, and exponentiation
Exponentiation is a mathematics, mathematical operation (mathematics), operation, written as , involving two numbers, the ''Base (exponentiation), base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". W ...

are three of the most fundamental arithmetic operations. The inverse of addition is subtraction, and the inverse of multiplication is division
Division or divider may refer to:
Mathematics
*Division (mathematics)
Division is one of the four basic operations of arithmetic, the ways that numbers are combined to make new numbers. The other operations are addition, subtraction, and ...

. Similarly, a logarithm is the inverse operation of exponentiation
Exponentiation is a mathematics, mathematical operation (mathematics), operation, written as , involving two numbers, the ''Base (exponentiation), base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". W ...

. Exponentiation is when a number , the ''base'', is raised to a certain power , the ''exponent'', to give a value ; this is denoted
: $b^y=x.$
For example, raising to the power of gives : $2^3\; =\; 8$
The logarithm of base is the inverse operation, that provides the output from the input . That is, $y\; =\; \backslash log\_b\; x$ is equivalent to $x=b^y$ if is a positive real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...

. (If is not a positive real number, both exponentiation and logarithm can be defined but may take several values, which makes definitions much more complicated.)
One of the main historical motivations of introducing logarithms is the formula
:$\backslash log\_b(xy)=\backslash log\_b\; x\; +\; \backslash log\_b\; y,$
which allowed (before the invention of computers) reducing computation of multiplications and divisions to additions, subtractions and logarithm table looking.
Definition

Given a positivereal number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...

such that , the ''logarithm'' of a positive real number with respect to base is the exponent by which must be raised to yield . In other words, the logarithm of to base is the unique real number such that $b^y\; =\; x$.
The logarithm is denoted "" (pronounced as "the logarithm of to base ", "the logarithm of ", or most commonly "the log, base , of ").
An equivalent and more succinct definition is that the function is the inverse function
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

to the function $x\backslash mapsto\; b^x$.
Examples

* , since . * Logarithms can also be negative: $\backslash log\_2\; \backslash !\; \backslash frac\; =\; -1$ since $2^\; =\; \backslash frac\; =\; \backslash frac.$ * is approximately 2.176, which lies between 2 and 3, just as 150 lies between and . * For any base , and , since and , respectively.Logarithmic identities

Several important formulas, sometimes called ''logarithmic identities'' or ''logarithmic laws'', relate logarithms to one another.Product, quotient, power, and root

The logarithm of a product is the sum of the logarithms of the numbers being multiplied; the logarithm of the ratio of two numbers is the difference of the logarithms. The logarithm of the -th power of a number is '' ''times the logarithm of the number itself; the logarithm of a -th root is the logarithm of the number divided by . The following table lists these identities with examples. Each of the identities can be derived after substitution of the logarithm definitions $x\; =\; b^$ or $y\; =\; b^$ in the left hand sides.Change of base

The logarithm can be computed from the logarithms of and with respect to an arbitrary base using the following formula: :$\backslash log\_b\; x\; =\; \backslash frac.\backslash ,$ Starting from the defining identity : $x\; =\; b^$ we can apply to both sides of this equation, to get : $\backslash log\_k\; x\; =\; \backslash log\_k\; \backslash left(b^\backslash right)\; =\; \backslash log\_b\; x\; \backslash cdot\; \backslash log\_k\; b$. Solving for $\backslash log\_b\; x$ yields: : $\backslash log\_b\; x\; =\; \backslash frac$, showing the conversion factor from given $\backslash log\_k$-values to their corresponding $\backslash log\_b$-values to be $(\backslash log\_k\; b)^.$ Typical scientific calculators calculate the logarithms to bases 10 and . Logarithms with respect to any base can be determined using either of these two logarithms by the previous formula: :$\backslash log\_b\; x\; =\; \backslash frac\; =\; \backslash frac.$ Given a number and its logarithm to an unknown base , the base is given by: :$b\; =\; x^\backslash frac,$ which can be seen from taking the defining equation $x\; =\; b^\; =\; b^y$ to the power of $\backslash tfrac.$Particular bases

Among all choices for the base, three are particularly common. These are , (theirrational
Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...

mathematical constant ≈ 2.71828), and (the binary logarithm
In mathematics, the binary logarithm () is the exponentiation, power to which the number must be exponentiation, raised to obtain the value . That is, for any real number ,
:x=\log_2 n \quad\Longleftrightarrow\quad 2^x=n.
For example, the ...

). In mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...

, the logarithm base is widespread because of analytical properties explained below. On the other hand, logarithms are easy to use for manual calculations in the decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...

number system:
:$\backslash log\_(10\; x)\; =\; \backslash log\_\; 10\; +\; \backslash log\_\; x\; =\; 1\; +\; \backslash log\_\; x.\backslash $
Thus, is related to the number of decimal digit
A numerical digit (often shortened to just digit) is a single symbol used alone (such as "2") or in combinations (such as "25"), to represent numbers in a Positional notation, positional numeral system. The name "digit" comes from the fact that t ...

s of a positive integer : the number of digits is the smallest integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...

strictly bigger than . For example, is approximately 3.15. The next integer is 4, which is the number of digits of 1430. Both the natural logarithm and the logarithm to base two are used in information theory
Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of information. The field was originally established by the works of Harry Nyquist a ...

, corresponding to the use of nats or bits as the fundamental units of information, respectively. Binary logarithms are also used in computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...

, where the binary system
A binary system is a system of two astronomical bodies which are close enough that their gravitational attraction causes them to orbit each other around a barycenter ''(also see Barycenter#Gallery, animated examples)''. More restrictive definitio ...

is ubiquitous; in music theory
Music theory is the study of the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory". The first is the "Elements of music, rudiments", that are needed to understand ...

, where a pitch ratio of two (the octave
In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the Pythagorean interval, diapason) is the interval (music), interval between one musical Pitch (music), pitch and another with double its frequency. The octave rela ...

) is ubiquitous and the number of cents between any two pitches is the binary logarithm, times 1200, of their ratio (that is, 100 cents per equal-temperament semitone
A semitone, also called a half step or a half tone, is the smallest interval (music), musical interval commonly used in Western tonal music, and it is considered the most Consonance and dissonance#Dissonance, dissonant when sounded harmonically ...

); and in photography
Photography is the visual art, art, application, and practice of creating durable images by recording light, either electronically by means of an image sensor, or chemically by means of a light-sensitive material such as photographic film. It i ...

to measure exposure value
In photography, exposure value (EV) is a number that represents a combination of a camera's shutter speed and f-number, such that all combinations that yield the same exposure (photography), exposure have the same EV (for any fixed scene luminanc ...

s, light levels, exposure times, aperture
In optics, an aperture is a hole or an opening through which light travels. More specifically, the aperture and focal length of an optical system determine the cone angle of a bundle of ray (optics), rays that come to a focus (optics), focus ...

s, and film speed
Film speed is the measure of a photographic film's photosensitivity, sensitivity to light, determined by sensitometry and measured on #Film, various numerical scales, the most recent being the #ISO, ISO system. A closely related ISO system is us ...

s in "stops".
The following table lists common notations for logarithms to these bases and the fields where they are used. Many disciplines write instead of , when the intended base can be determined from the context. The notation also occurs. The "ISO notation" column lists designations suggested by the International Organization for Standardization
The International Organization for Standardization (ISO ) is an international standard development organization composed of representatives from the national standards organizations of member countries. Membership requirements are given in Art ...

(ISO 80000-2
ISO 80000 or IEC 80000 is an international standard
An international standard is a technical standard developed by one or more international standards organization, standards organizations. International standards are available for consideratio ...

). Because the notation has been used for all three bases (or when the base is indeterminate or immaterial), the intended base must often be inferred based on context or discipline. In computer science, usually refers to , and in mathematics usually refers to . In other contexts, often means .
History

The history of logarithms in seventeenth-century Europe is the discovery of a new function that extended the realm of analysis beyond the scope of algebraic methods. The method of logarithms was publicly propounded byJohn Napier
John Napier of Merchiston (; 1 February 1550 – 4 April 1617), nicknamed Marvellous Merchiston, was a Scottish landowner known as a mathematician, physicist, and astronomer. He was the 8th Laird of Merchiston. His Latinisation of names, L ...

in 1614, in a book titled '' Mirifici Logarithmorum Canonis Descriptio'' (''Description of the Wonderful Rule of Logarithms''). Prior to Napier's invention, there had been other techniques of similar scopes, such as the prosthaphaeresis
Prosthaphaeresis (from the Greek ''προσθαφαίρεσις'') was an algorithm used in the late 16th century and early 17th century for approximate multiplication and Division (mathematics), division using formulas from trigonometry. For the ...

or the use of tables of progressions, extensively developed by Jost Bürgi around 1600. Napier coined the term for logarithm in Middle Latin, “logarithmus,” derived from the Greek, literally meaning, “ratio-number,” from ''logos'' “proportion, ratio, word” + ''arithmos'' “number”.
The common logarithm
In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs (mathematician), Henry Briggs, an English ...

of a number is the index of that power of ten which equals the number. Speaking of a number as requiring so many figures is a rough allusion to common logarithm, and was referred to by Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, ...

as the “order of a number”. The first real logarithms were heuristic methods to turn multiplication into addition, thus facilitating rapid computation. Some of these methods used tables derived from trigonometric identities. Such methods are called prosthaphaeresis
Prosthaphaeresis (from the Greek ''προσθαφαίρεσις'') was an algorithm used in the late 16th century and early 17th century for approximate multiplication and Division (mathematics), division using formulas from trigonometry. For the ...

.
Invention of the function now known as the natural logarithm
The natural logarithm of a number is its logarithm to the base (exponentiation), base of the mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approximately equal to . The natur ...

began as an attempt to perform a quadrature of a rectangular hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A h ...

by Grégoire de Saint-Vincent, a Belgian Jesuit residing in Prague. Archimedes had written ''The Quadrature of the Parabola
''Quadrature of the Parabola'' ( el, Τετραγωνισμὸς παραβολῆς) is a treatise on geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the ...

'' in the third century BC, but a quadrature for the hyperbola eluded all efforts until Saint-Vincent published his results in 1647. The relation that the logarithm provides between a geometric progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For ex ...

in its argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called a conclusion. Arguments can be studied from three main perspectives: the logical, the dialect ...

and an arithmetic progression
An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...

of values, prompted A. A. de Sarasa to make the connection of Saint-Vincent's quadrature and the tradition of logarithms in prosthaphaeresis
Prosthaphaeresis (from the Greek ''προσθαφαίρεσις'') was an algorithm used in the late 16th century and early 17th century for approximate multiplication and Division (mathematics), division using formulas from trigonometry. For the ...

, leading to the term “hyperbolic logarithm”, a synonym for natural logarithm. Soon the new function was appreciated by Christiaan Huygens
Christiaan Huygens, Lord of Zeelhem, ( , , ; also spelled Huyghens; la, Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typica ...

, and James Gregory. The notation Log y was adopted by Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...

in 1675, and the next year he connected it to the integral
In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...

$\backslash int\; \backslash frac\; .$
Before Euler developed his modern conception of complex natural logarithms, Roger Cotes
Roger Cotes (10 July 1682 – 5 June 1716) was an English mathematician, known for working closely with Isaac Newton by proofreading the second edition of his famous book, the ''Philosophiae Naturalis Principia Mathematica, Principia'', before ...

had a nearly equivalent result when he showed in 1714 that
:$\backslash log(\backslash cos\; \backslash theta\; +\; i\backslash sin\; \backslash theta)\; =\; i\backslash theta$.
Logarithm tables, slide rules, and historical applications

By simplifying difficult calculations before calculators and computers became available, logarithms contributed to the advance of science, especiallyastronomy
Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...

. They were critical to advances in surveying
Surveying or land surveying is the technique, profession, art, and science of determining the land, terrestrial Two-dimensional space#In geometry, two-dimensional or Three-dimensional space#In Euclidean geometry, three-dimensional positions of ...

, celestial navigation
Celestial navigation, also known as astronavigation, is the practice of position fixing using stars and other celestial bodies that enables a navigator to accurately determine their actual current physical position in space (or on the surface of ...

, and other domains. Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...

called logarithms
::"... admirable artifice which, by reducing to a few days the labour of many months, doubles the life of the astronomer, and spares him the errors and disgust inseparable from long calculations."
As the function is the inverse function of , it has been called an antilogarithm. Nowadays, this function is more commonly called an exponential function
The exponential function is a mathematical Function (mathematics), function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponentiation, exponent). Unless otherwise specified, the term generally refers to the positiv ...

.
Log tables

A key tool that enabled the practical use of logarithms was the '' table of logarithms''. The first such table was compiled by Henry Briggs in 1617, immediately after Napier's invention but with the innovation of using 10 as the base. Briggs' first table contained thecommon logarithm
In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs (mathematician), Henry Briggs, an English ...

s of all integers in the range from 1 to 1000, with a precision of 14 digits. Subsequently, tables with increasing scope were written. These tables listed the values of for any number in a certain range, at a certain precision. Base-10 logarithms were universally used for computation, hence the name common logarithm, since numbers that differ by factors of 10 have logarithms that differ by integers. The common logarithm of can be separated into an integer part and a fractional part
The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. If the latter is defined as the largest integer not greater than , called floor function, floor of or \lfloor x\rfloor, its frac ...

, known as the characteristic and mantissa. Tables of logarithms need only include the mantissa, as the characteristic can be easily determined by counting digits from the decimal point. The characteristic of is one plus the characteristic of , and their mantissas are the same. Thus using a three-digit log table, the logarithm of 3542 is approximated by
:$\backslash log\_3542\; =\; \backslash log\_(1000\; \backslash cdot\; 3.542)\; =\; 3\; +\; \backslash log\_3.542\; \backslash approx\; 3\; +\; \backslash log\_3.54\; \backslash ,$
Greater accuracy can be obtained by interpolation
In the mathematics, mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points.
In engineering and science, one ...

:
:$\backslash log\_3542\; \backslash approx\; 3\; +\; \backslash log\_3.54\; +\; 0.2\; (\backslash log\_3.55-\backslash log\_3.54)\backslash ,$
The value of can be determined by reverse look up in the same table, since the logarithm is a monotonic function
In mathematics, a monotonic function (or monotone function) is a function (mathematics), function between List of order structures in mathematics, ordered sets that preserves or reverses the given order relation, order. This concept first aro ...

.
Computations

The product and quotient of two positive numbers and ' were routinely calculated as the sum and difference of their logarithms. The product or quotient came from looking up the antilogarithm of the sum or difference, via the same table: :$cd\; =\; 10^\; \backslash ,\; 10^\; =\; 10^$ and :$\backslash frac\; c\; d\; =\; c\; d^\; =\; 10^.$ For manual calculations that demand any appreciable precision, performing the lookups of the two logarithms, calculating their sum or difference, and looking up the antilogarithm is much faster than performing the multiplication by earlier methods such astrigonometric identities
In trigonometry, trigonometric identities are Equality (mathematics), equalities that involve trigonometric functions and are true for every value of the occurring Variable (mathematics), variables for which both sides of the equality are defined. ...

.
Calculations of powers and roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* The Root (magazine), ''The Root'' (magazine), an onli ...

are reduced to multiplications or divisions and lookups by
:$c^d\; =\; \backslash left(10^\backslash right)^d\; =\; 10^$
and
:$\backslash sqrt;\; href="/html/ALL/s/.html"\; ;"title="">$
Trigonometric calculations were facilitated by tables that contained the common logarithms of trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...

s.
Slide rules

Another critical application was theslide rule
The slide rule is a mechanical analog computer which is used primarily for multiplication and division (mathematics), division, and for functions such as exponents, Nth root, roots, logarithms, and trigonometry. It is not typically designed for ...

, a pair of logarithmically divided scales used for calculation. The non-sliding logarithmic scale, Gunter's rule, was invented shortly after Napier's invention. William Oughtred
William Oughtred ( ; 5 March 1574 – 30 June 1660), also Owtred, Uhtred, etc., was an Kingdom of England, English mathematician and Anglican ministry, Anglican clergyman.'Oughtred (William)', in P. Bayle, translated and revised by J.P. Bernar ...

enhanced it to create the slide rule—a pair of logarithmic scales movable with respect to each other. Numbers are placed on sliding scales at distances proportional to the differences between their logarithms. Sliding the upper scale appropriately amounts to mechanically adding logarithms, as illustrated here:
For example, adding the distance from 1 to 2 on the lower scale to the distance from 1 to 3 on the upper scale yields a product of 6, which is read off at the lower part. The slide rule was an essential calculating tool for engineers and scientists until the 1970s, because it allows, at the expense of precision, much faster computation than techniques based on tables.
Analytic properties

A deeper study of logarithms requires the concept of a '' function''. A function is a rule that, given one number, produces another number. An example is the function producing the -th power of from any real number , where the base is a fixed number. This function is written as . When is positive and unequal to 1, we show below that is invertible when considered as a function from the reals to the positive reals.Existence

Let be a positive real number not equal to 1 and let . It is a standard result in real analysis that any continuous strictly monotonic function is bijective between its domain and range. This fact follows from theintermediate value theorem
In mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, in ...

., section III.3 Now, is strictly increasing
In mathematics, a monotonic function (or monotone function) is a function (mathematics), function between List of order structures in mathematics, ordered sets that preserves or reverses the given order relation, order. This concept first aro ...

(for ), or strictly decreasing (for ), is continuous, has domain $\backslash R$, and has range $\backslash R\_$. Therefore, is a bijection from $\backslash R$ to $\backslash R\_$. In other words, for each positive real number , there is exactly one real number such that $b^x\; =\; y$.
We let $\backslash log\_b\backslash colon\backslash R\_\backslash to\backslash R$ denote the inverse of . That is, is the unique real number such that $b^x\; =\; y$. This function is called the base- ''logarithm function'' or ''logarithmic function'' (or just ''logarithm'').
Characterization by the product formula

The function can also be essentially characterized by the product formula :$\backslash log\_b(xy)\; =\; \backslash log\_b\; x\; +\; \backslash log\_b\; y.$ More precisely, the logarithm to any base is the onlyincreasing function
In mathematics, a monotonic function (or monotone function) is a function (mathematics), function between List of order structures in mathematics, ordered sets that preserves or reverses the given order relation, order. This concept first aro ...

''f'' from the positive reals to the reals satisfying and
:$f(xy)=f(x)+f(y).$
Graph of the logarithm function

As discussed above, the function is the inverse to the exponential function $x\backslash mapsto\; b^x$. Therefore, Their graphs correspond to each other upon exchanging the - and the -coordinates (or upon reflection at the diagonal line ), as shown at the right: a point on the graph of yields a point on the graph of the logarithm and vice versa. As a consequence, diverges to infinity (gets bigger than any given number) if grows to infinity, provided that is greater than one. In that case, is anincreasing function
In mathematics, a monotonic function (or monotone function) is a function (mathematics), function between List of order structures in mathematics, ordered sets that preserves or reverses the given order relation, order. This concept first aro ...

. For , tends to minus infinity instead. When approaches zero, goes to minus infinity for (plus infinity for , respectively).
Derivative and antiderivative

Analytic properties of functions pass to their inverses. Thus, as is a continuous anddifferentiable function
In mathematics, a differentiable function of one Real number, real variable is a Function (mathematics), function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differ ...

, so is . Roughly, a continuous function is differentiable if its graph has no sharp "corners". Moreover, as the derivative
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...

of evaluates to by the properties of the exponential function
The exponential function is a mathematical Function (mathematics), function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponentiation, exponent). Unless otherwise specified, the term generally refers to the positiv ...

, the chain rule
In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x) ...

implies that the derivative of is given by
: $\backslash frac\; \backslash log\_b\; x\; =\; \backslash frac.$
That is, the slope
In mathematics, the slope or gradient of a line
Line most often refers to:
* Line (geometry)
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional object ...

of the tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given Point (geometry), point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitesimal, infinitely ...

touching the graph of the logarithm at the point equals .
The derivative of is ; this implies that is the unique antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function (mathematics), function is a differentiable function whose derivative is equal to the original function . This ca ...

of that has the value 0 for . It is this very simple formula that motivated to qualify as "natural" the natural logarithm; this is also one of the main reasons of the importance of the constant .
The derivative with a generalized functional argument is
:$\backslash frac\; \backslash ln\; f(x)\; =\; \backslash frac.$
The quotient at the right hand side is called the logarithmic derivative
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...

of '. Computing by means of the derivative of is known as logarithmic differentiation
In calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study ...

. The antiderivative of the natural logarithm
The natural logarithm of a number is its logarithm to the base (exponentiation), base of the mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approximately equal to . The natur ...

is:
: $\backslash int\; \backslash ln(x)\; \backslash ,dx\; =\; x\; \backslash ln(x)\; -\; x\; +\; C.$
Related formulas, such as antiderivatives of logarithms to other bases can be derived from this equation using the change of bases.
Integral representation of the natural logarithm

Thedefinite integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...

:
:$\backslash ln\; t\; =\; \backslash int\_1^t\; \backslash frac\; \backslash ,\; dx.$
This definition has the advantage that it does not rely on the exponential function or any trigonometric functions; the definition is in terms of an integral of a simple reciprocal. As an integral, equals the area between the -axis and the graph of the function , ranging from to . This is a consequence of the fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, ...

and the fact that the derivative of is . Product and power logarithm formulas can be derived from this definition. For example, the product formula is deduced as:
:$\backslash ln(tu)\; =\; \backslash int\_1^\; \backslash frac\; \backslash ,\; dx\; \backslash \; \backslash stackrel\; =\; \backslash int\_1^\; \backslash frac\; \backslash ,\; dx\; +\; \backslash int\_t^\; \backslash frac\; \backslash ,\; dx\; \backslash \; \backslash stackrel\; =\; \backslash ln(t)\; +\; \backslash int\_1^u\; \backslash frac\; \backslash ,\; dw\; =\; \backslash ln(t)\; +\; \backslash ln(u).$
The equality (1) splits the integral into two parts, while the equality (2) is a change of variable (). In the illustration below, the splitting corresponds to dividing the area into the yellow and blue parts. Rescaling the left hand blue area vertically by the factor and shrinking it by the same factor horizontally does not change its size. Moving it appropriately, the area fits the graph of the function again. Therefore, the left hand blue area, which is the integral of from to is the same as the integral from 1 to . This justifies the equality (2) with a more geometric proof.
The power formula may be derived in a similar way:
:$\backslash ln(t^r)\; =\; \backslash int\_1^\; \backslash fracdx\; =\; \backslash int\_1^t\; \backslash frac\; \backslash left(rw^\; \backslash ,\; dw\backslash right)\; =\; r\; \backslash int\_1^t\; \backslash frac\; \backslash ,\; dw\; =\; r\; \backslash ln(t).$
The second equality uses a change of variables (integration by substitution
In calculus, integration by substitution, also known as ''u''-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for derivative, differentiati ...

), .
The sum over the reciprocals of natural numbers,
:$1\; +\; \backslash frac\; 1\; 2\; +\; \backslash frac\; 1\; 3\; +\; \backslash cdots\; +\; \backslash frac\; 1\; n\; =\; \backslash sum\_^n\; \backslash frac,$
is called the harmonic series. It is closely tied to the infinity
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Since the time of the Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophical nature of infinit ...

, the difference,
:$\backslash sum\_^n\; \backslash frac\; -\; \backslash ln(n),$
converges (i.e. gets arbitrarily close) to a number known as the Euler–Mascheroni constant
Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma (letter), gamma ().
It is defined as the limit of a sequence, limiting difference between th ...

. This relation aids in analyzing the performance of algorithms such as quicksort
Quicksort is an efficient, general-purpose sorting algorithm. Quicksort was developed by British computer scientist Tony Hoare in 1959 and published in 1961, it is still a commonly used algorithm for sorting. Overall, it is slightly faster than ...

.
Transcendence of the logarithm

Real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...

s that are not algebraic are called transcendental; for example, and '' e'' are such numbers, but $\backslash sqrt$ is not. Almost all
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...

real numbers are transcendental. The logarithm is an example of a transcendental function
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...

. The Gelfond–Schneider theorem asserts that logarithms usually take transcendental, i.e. "difficult" values.
Calculation

Logarithms are easy to compute in some cases, such as . In general, logarithms can be calculated usingpower series
In mathematics, a power series (in one variable (mathematics), variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th te ...

or the arithmetic–geometric mean, or be retrieved from a precalculated logarithm table that provides a fixed precision. Newton's method
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a re ...

, an iterative method to solve equations approximately, can also be used to calculate the logarithm, because its inverse function, the exponential function, can be computed efficiently. Using look-up tables, CORDIC
CORDIC (for "coordinate rotation digital computer"), also known as Volder's algorithm, or: Digit-by-digit method Circular CORDIC (Jack E. Volder), Linear CORDIC, Hyperbolic CORDIC (John Stephen Walther), and Generalized Hyperbolic CORDIC (GH C ...

-like methods can be used to compute logarithms by using only the operations of addition and bit shifts. Moreover, the binary logarithm algorithm calculates recursively
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...

, based on repeated squarings of , taking advantage of the relation
:$\backslash log\_2\backslash left(x^2\backslash right)\; =\; 2\; \backslash log\_2\; ,\; x,\; .$
Power series

Taylor series

For any real number that satisfies , the following formula holds: :$\backslash begin\backslash ln\; (z)\; \&=\; \backslash frac\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; +\; \backslash cdots\; \backslash \backslash \; \&=\; \backslash sum\_^\backslash infty\; (-1)^\backslash frac\; \backslash end$ This is a shorthand for saying that can be approximated to a more and more accurate value by the following expressions: :$\backslash begin\; (z-1)\; \&\; \&\; \backslash \backslash \; (z-1)\; \&\; -\; \&\; \backslash frac\; \&\; \backslash \backslash \; (z-1)\; \&\; -\; \&\; \backslash frac\; \&\; +\; \&\; \backslash frac\; \backslash \backslash \; \backslash vdots\; \&\; \backslash end$ For example, with the third approximation yields 0.4167, which is about 0.011 greater than . Thisseries
Series may refer to:
People with the name
* Caroline Series
Caroline Mary Series (born 24 March 1951) is an English mathematician known for her work in hyperbolic geometry, Kleinian groups and dynamical systems.
Early life and education
S ...

approximates with arbitrary precision, provided the number of summands is large enough. In elementary calculus, is therefore the limit of this series. It is the Taylor series
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

of the Inverse hyperbolic tangent

Another series is based on the inverse hyperbolic tangent function: :$\backslash ln\; (z)\; =\; 2\backslash cdot\backslash operatorname\backslash ,\backslash frac\; =\; 2\; \backslash left\; (\; \backslash frac\; +\; \backslash frac^3\; +\; \backslash frac^5\; +\; \backslash cdots\; \backslash right\; ),$ for any real number . Usingsigma notation
In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: function (mathematics), fu ...

, this is also written as
:$\backslash ln\; (z)\; =\; 2\backslash sum\_^\backslash infty\backslash frac\backslash left(\backslash frac\backslash right)^.$
This series can be derived from the above Taylor series. It converges quicker than the Taylor series, especially if is close to 1. For example, for , the first three terms of the second series approximate with an error of about . The quick convergence for close to 1 can be taken advantage of in the following way: given a low-accuracy approximation and putting
:$A\; =\; \backslash frac\; z,$
the logarithm of is:
:$\backslash ln\; (z)=y+\backslash ln\; (A).$
The better the initial approximation is, the closer is to 1, so its logarithm can be calculated efficiently. can be calculated using the exponential series, which converges quickly provided is not too large. Calculating the logarithm of larger can be reduced to smaller values of by writing , so that .
A closely related method can be used to compute the logarithm of integers. Putting $\backslash textstyle\; z=\backslash frac$ in the above series, it follows that:
:$\backslash ln\; (n+1)\; =\; \backslash ln(n)\; +\; 2\backslash sum\_^\backslash infty\backslash frac\backslash left(\backslash frac\backslash right)^.$
If the logarithm of a large integer is known, then this series yields a fast converging series for , with a rate of convergence
In numerical analysis, the order of convergence and the rate of convergence of a limit of a sequence, convergent sequence are quantities that represent how quickly the sequence approaches its limit. A sequence (x_n) that converges to x^* is said ...

of $\backslash left(\backslash frac\backslash right)^$.
Arithmetic–geometric mean approximation

The arithmetic–geometric mean yields high precision approximations of theCarl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

):
:$\backslash ln\; (x)\; \backslash approx\; \backslash frac\; -\; m\; \backslash ln(2).$
Here denotes the arithmetic–geometric mean of and . It is obtained by repeatedly calculating the average (arithmetic mean
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

) and $\backslash sqrt$ (geometric mean
In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the Nt ...

) of and then let those two numbers become the next and . The two numbers quickly converge to a common limit which is the value of . is chosen such that
:$x\; \backslash ,2^m\; >\; 2^.\backslash ,$
to ensure the required precision. A larger makes the calculation take more steps (the initial and are farther apart so it takes more steps to converge) but gives more precision. The constants and can be calculated with quickly converging series.
Feynman's algorithm

While atLos Alamos National Laboratory
Los Alamos National Laboratory (often shortened as Los Alamos and LANL) is one of the sixteen research and development Laboratory, laboratories of the United States Department of Energy National Laboratories, United States Department of Energy ...

working on the Manhattan Project
The Manhattan Project was a research and development undertaking during World War II that produced the first nuclear weapons. It was led by the United States with the support of the United Kingdom and Canada. From 1942 to 1946, the project w ...

, Richard Feynman
Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superflu ...

developed a bit-processing algorithm, to compute the logarithm, that is similar to long division and was later used in the Connection Machine
A Connection Machine (CM) is a member of a series of massively parallel supercomputers that grew out of doctoral research on alternatives to the traditional von Neumann architecture of computers by Danny Hillis at Massachusetts Institute of Techno ...

. The algorithm uses the fact that every real number is representable as a product of distinct factors of the form . The algorithm sequentially builds that product , starting with and : if , then it changes to . It then increases $k$ by one regardless. The algorithm stops when is large enough to give the desired accuracy. Because is the sum of the terms of the form corresponding to those for which the factor was included in the product , may be computed by simple addition, using a table of for all . Any base may be used for the logarithm table.
Applications

Logarithms have many applications inside and outside mathematics. Some of these occurrences are related to the notion ofscale invariance
In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality.
The technical term ...

. For example, each chamber of the shell of a nautilus
The nautilus (, ) is a pelagic marine mollusc of the cephalopod family Nautilidae. The nautilus is the sole extant family of the superfamily Nautilaceae and of its smaller but near equal suborder, Nautilina.
It comprises six living species in t ...

is an approximate copy of the next one, scaled by a constant factor. This gives rise to a logarithmic spiral
A logarithmic spiral, equiangular spiral, or growth spiral is a self-similarity, self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewi ...

. Benford's law
Benford's law, also known as the Newcomb–Benford law, the law of anomalous numbers, or the first-digit law, is an observation that in many real-life sets of numerical data, the leading digit is likely to be small.Arno Berger and Theodore P ...

on the distribution of leading digits can also be explained by scale invariance. Logarithms are also linked to self-similarity
__NOTOC__
In mathematics, a self-similar object is exactly or approximately similarity (geometry), similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines ...

. For example, logarithms appear in the analysis of algorithms that solve a problem by dividing it into two similar smaller problems and patching their solutions. The dimensions of self-similar geometric shapes, that is, shapes whose parts resemble the overall picture are also based on logarithms. Logarithmic scale
A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers. Such a ...

s are useful for quantifying the relative change of a value as opposed to its absolute difference. Moreover, because the logarithmic function grows very slowly for large , logarithmic scales are used to compress large-scale scientific data. Logarithms also occur in numerous scientific formulas, such as the Tsiolkovsky rocket equation
Konstantin Eduardovich Tsiolkovsky (russian: Константи́н Эдуа́рдович Циолко́вский , , p=kənstɐnʲˈtʲin ɪdʊˈardəvʲɪtɕ tsɨɐlˈkofskʲɪj , a=Ru-Konstantin Tsiolkovsky.oga; – 19 September 1935) ...

, the Fenske equation
The Fenske equation in continuous fractional distillation
Fractional distillation is the separation process, separation of a mixture into its component parts, or fraction (chemistry), fractions. Chemical compounds are separated by heating them ...

, or the Nernst equation
In electrochemistry
Electrochemistry is the branch of physical chemistry concerned with the relationship between Electric potential, electrical potential difference, as a measurable and quantitative phenomenon, and identifiable chemical change, ...

.
Logarithmic scale

Scientific quantities are often expressed as logarithms of other quantities, using a ''logarithmic scale''. For example, thedecibel
The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a Power, root-power, and field quantities, power or root-power quantity on a logarithmic scale. Two signals whose ...

is a unit of measurement
A unit of measurement is a definite magnitude of a quantity
Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", ...

associated with logarithmic-scale quantities
Quantity or amount is a property that can exist as a Counting, multitude or Magnitude (mathematics), magnitude, which illustrate discontinuity (mathematics), discontinuity and continuum (theory), continuity. Quantities can be compared in terms o ...

. It is based on the common logarithm of ratio
In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...

s—10 times the common logarithm of a power
Power most often refers to:
* Power (physics)
In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equal to one joule per second. In older work ...

ratio or 20 times the common logarithm of a voltage
Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a Electrostatics, static electric field, it corresponds to the Work (electrical), w ...

ratio. It is used to quantify the loss of voltage levels in transmitting electrical signals, to describe power levels of sounds in acoustics
Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...

, and the absorbance
Absorbance is defined as "the logarithm of the ratio of incident to transmitted radiant power through a sample (excluding the effects on cell walls)". Alternatively, for samples which scatter light, absorbance may be defined as "the negative lo ...

of light in the fields of spectrometry and optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...

. The signal-to-noise ratio
Signal-to-noise ratio (SNR or S/N) is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to the noise power, often expressed in ...

describing the amount of unwanted noise
Noise is unwanted sound considered unpleasant, loud or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference arise ...

in relation to a (meaningful) signal
In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The ''IEEE Transactions on Signal Processing'' ...

is also measured in decibels. In a similar vein, the peak signal-to-noise ratio
Peak signal-to-noise ratio (PSNR) is an engineering term for the ratio between the maximum possible power of a Signal (information theory), signal and the power of corrupting noise that affects the fidelity of its representation. Because many sign ...

is commonly used to assess the quality of sound and image compression
Image compression is a type of data compression applied to digital images, to reduce their cost for computer data storage, storage or data transmission, transmission. Algorithms may take advantage of visual perception and the statistical properti ...

methods using the logarithm.
The strength of an earthquake is measured by taking the common logarithm of the energy emitted at the quake. This is used in the moment magnitude scale
The moment magnitude scale (MMS; denoted explicitly with or Mw, and generally implied with use of a single M for magnitude) is a measure of an earthquake's magnitude ("size" or strength) based on its seismic moment. It was defined in a 1979 pape ...

or the Richter magnitude scale
The Richter scale —also called the Richter magnitude scale, Richter's magnitude scale, and the Gutenberg–Richter scale—is a measure of the strength of earthquakes, developed by Charles Francis Richter and presented in his landmark 1935 p ...

. For example, a 5.0 earthquake releases 32 times and a 6.0 releases 1000 times the energy of a 4.0. Apparent magnitude
Apparent magnitude () is a measure of the Irradiance, brightness of a star or other astronomical object observed from Earth. An object's apparent magnitude depends on its intrinsic luminosity, its distance from Earth, and any extinction (astron ...

measures the brightness of stars logarithmically. In chemistry
Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, properties ...

the negative of the decimal logarithm, the decimal , is indicated by the letter p. For instance, pH is the decimal cologarithm of the activity of hydronium
In chemistry, hydronium (hydroxonium in traditional British English) is the common name for the aqueous cation , the type of oxonium ion produced by protonation of water. It is often viewed as the positive ion present when an Arrhenius acid is di ...

ions (the form hydrogen
Hydrogen is the chemical element with the Symbol (chemistry), symbol H and atomic number 1. Hydrogen is the lightest element. At standard temperature and pressure, standard conditions hydrogen is a gas of diatomic molecules having the chemical ...

ion
An ion () is an atom
Every atom is composed of a atomic nucleus, nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has n ...

s take in water). The activity of hydronium ions in neutral water is 10slope
In mathematics, the slope or gradient of a line
Line most often refers to:
* Line (geometry)
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional object ...

equal to the logarithm of . Log-log graphs scale both axes logarithmically, which causes functions of the form to be depicted as straight lines with slope equal to the exponent . This is applied in visualizing and analyzing power law
In statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, a ...

s.
Psychology

Logarithms occur in several laws describing human perception: Hick's law proposes a logarithmic relation between the time individuals take to choose an alternative and the number of choices they have.Fitts's law
Fitts's law (often cited as Fitts' law) is a predictive model of human movement primarily used in human–computer interaction and Human factors and ergonomics, ergonomics. The scientific law, law predicts that the time required to rapidly move to ...

predicts that the time required to rapidly move to a target area is a logarithmic function of the distance to and the size of the target. In psychophysics
Psychophysics quantitatively investigates the relationship between physical stimulus (physiology), stimuli and the sensation (psychology), sensations and perceptions they produce. Psychophysics has been described as "the scientific study of the ...

, the Weber–Fechner law
The Weber–Fechner laws are two related hypothesis, hypotheses in the field of psychophysics, known as Weber's law and Fechner's law. Both scientific law, laws relate to human perception, more specifically the relation between the actual change ...

proposes a logarithmic relationship between stimulus and sensation
Sensation (psychology) refers to the processing of the senses by the sensory system.
Sensation or sensations may also refer to:
In arts and entertainment In literature
*Sensation (fiction), a fiction writing mode
*Sensation novel, a British ...

such as the actual vs. the perceived weight of an item a person is carrying. (This "law", however, is less realistic than more recent models, such as Stevens's power law
Stevens' power law is an empirical relationship in psychophysics
Psychophysics quantitatively investigates the relationship between physical stimulus (physiology), stimuli and the sensation (psychology), sensations and perceptions they produc ...

.)
Psychological studies found that individuals with little mathematics education tend to estimate quantities logarithmically, that is, they position a number on an unmarked line according to its logarithm, so that 10 is positioned as close to 100 as 100 is to 1000. Increasing education shifts this to a linear estimate (positioning 1000 10 times as far away) in some circumstances, while logarithms are used when the numbers to be plotted are difficult to plot linearly.
Probability theory and statistics

Logarithms arise inprobability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

: the law of large numbers
In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials shou ...

dictates that, for a fair coin
In probability theory and statistics, a sequence of Independence (probability theory), independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin. One for which the probability is not 1/2 is cal ...

, as the number of coin-tosses increases to infinity, the observed proportion of heads approaches one-half. The fluctuations of this proportion about one-half are described by the law of the iterated logarithm.
Logarithms also occur in log-normal distribution
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed, ...

s. When the logarithm of a random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...

has a normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real number, real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The param ...

, the variable is said to have a log-normal distribution. Log-normal distributions are encountered in many fields, wherever a variable is formed as the product of many independent positive random variables, for example in the study of turbulence.
Logarithms are used for maximum-likelihood estimation
In statistics, maximum likelihood estimation (MLE) is a method of estimation theory, estimating the Statistical parameter, parameters of an assumed probability distribution, given some observed data. This is achieved by Mathematical optimization, ...

of parametric statistical model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of Sample (statistics), sample data (and similar data from a larger Statistical population, population). A statistical model repres ...

s. For such a model, the likelihood function
The likelihood function (often simply called the likelihood) represents the probability of Realization (probability), random variable realizations conditional on particular values of the statistical parameters. Thus, when evaluated on a Sample (st ...

depends on at least one parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...

that must be estimated. A maximum of the likelihood function occurs at the same parameter-value as a maximum of the logarithm of the likelihood (the "''log likelihood''"), because the logarithm is an increasing function. The log-likelihood is easier to maximize, especially for the multiplied likelihoods for independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independen ...

random variables.
Benford's law
Benford's law, also known as the Newcomb–Benford law, the law of anomalous numbers, or the first-digit law, is an observation that in many real-life sets of numerical data, the leading digit is likely to be small.Arno Berger and Theodore P ...

describes the occurrence of digits in many data set A data set (or dataset) is a collection of data. In the case of tabular data, a data set corresponds to one or more table (database), database tables, where every column (database), column of a table represents a particular Variable (computer scienc ...

s, such as heights of buildings. According to Benford's law, the probability that the first decimal-digit of an item in the data sample is (from 1 to 9) equals , ''regardless'' of the unit of measurement. Thus, about 30% of the data can be expected to have 1 as first digit, 18% start with 2, etc. Auditors examine deviations from Benford's law to detect fraudulent accounting.
The logarithm transformation is a type of data transformation used to bring the empirical distribution closer to the assumed one.
Computational complexity

Analysis of algorithms
In computer science, the analysis of algorithms is the process of finding the computational complexity of algorithms—the amount of time, storage, or other resources needed to execute them. Usually, this involves determining a Function (mathem ...

is a branch of computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...

that studies the performance
A performance is an act of staging or presenting a play, concert, or other form of entertainment. It is also defined as the action or process of carrying out or accomplishing an action, task, or function.
Management science
In the work place ...

of algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...

s (computer programs solving a certain problem)., pp. 1–2 Logarithms are valuable for describing algorithms that divide a problem into smaller ones, and join the solutions of the subproblems.
For example, to find a number in a sorted list, the binary search algorithm
In computer science, binary search, also known as half-interval search, logarithmic search, or binary chop, is a search algorithm that finds the position of a target value within a sorted array. Binary search compares the target value to the m ...

checks the middle entry and proceeds with the half before or after the middle entry if the number is still not found. This algorithm requires, on average, comparisons, where is the list's length. Similarly, the merge sort
In computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied scie ...

algorithm sorts an unsorted list by dividing the list into halves and sorting these first before merging the results. Merge sort algorithms typically require a time approximately proportional to . The base of the logarithm is not specified here, because the result only changes by a constant factor when another base is used. A constant factor is usually disregarded in the analysis of algorithms under the standard uniform cost model.
A function is said to grow logarithmically if is (exactly or approximately) proportional to the logarithm of . (Biological descriptions of organism growth, however, use this term for an exponential function.) For example, any natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...

can be represented in binary form
Binary form is a musical form in 2 related sections, both of which are usually repeated. Binary is also a structure used to choreograph dance. In music this is usually performed as A-A-B-B.
Binary form was popular during the Baroque music, Baroq ...

in no more than bits. In other words, the amount of memory
Memory is the faculty of the mind by which data or information is Encoding (memory), encoded, stored, and retrieved when needed. It is the retention of information over time for the purpose of influencing future action. If Foresight (psycholo ...

needed to store grows logarithmically with .
Entropy and chaos

Entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...

is broadly a measure of the disorder of some system. In statistical thermodynamics
In physics, statistical mechanics is a mathematical framework that applies Statistics, statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the ma ...

, the entropy ''S'' of some physical system is defined as
:$S\; =\; -\; k\; \backslash sum\_i\; p\_i\; \backslash ln(p\_i).\backslash ,$
The sum is over all possible states of the system in question, such as the positions of gas particles in a container. Moreover, is the probability that the state is attained and is the Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas ...

. Similarly, entropy in information theory measures the quantity of information. If a message recipient may expect any one of possible messages with equal likelihood, then the amount of information conveyed by any one such message is quantified as bits.
Lyapunov exponents use logarithms to gauge the degree of chaoticity of a dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...

. For example, for a particle moving on an oval billiard table, even small changes of the initial conditions result in very different paths of the particle. Such systems are chaotic in a deterministic
Determinism is a Philosophy, philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motive ...

way, because small measurement errors of the initial state predictably lead to largely different final states. At least one Lyapunov exponent of a deterministically chaotic system is positive.
Fractals

Logarithms occur in definitions of thedimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...

of fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illus ...

s. Fractals are geometric objects that are self-similar in the sense that small parts reproduce, at least roughly, the entire global structure. The Sierpinski triangle (pictured) can be covered by three copies of itself, each having sides half the original length. This makes the Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point (geometry), po ...

of this structure . Another logarithm-based notion of dimension is obtained by counting the number of boxes needed to cover the fractal in question.
Music

Logarithms are related to musical tones and intervals. Inequal temperament
An equal temperament is a musical temperament or Musical tuning#Tuning systems, tuning system, which approximates Just intonation, just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequency, ...

, the frequency ratio depends only on the interval between two tones, not on the specific frequency, or pitch
Pitch may refer to:
Acoustic frequency
* Pitch (music), the perceived frequency of sound including "definite pitch" and "indefinite pitch"
** Absolute pitch or "perfect pitch"
** Pitch class, a set of all pitches that are a whole number of octave ...

, of the individual tones. For example, the note ''A'' has a frequency of 440 Hz and ''B-flat'' has a frequency of 466 Hz. The interval between ''A'' and ''B-flat'' is a semitone
A semitone, also called a half step or a half tone, is the smallest interval (music), musical interval commonly used in Western tonal music, and it is considered the most Consonance and dissonance#Dissonance, dissonant when sounded harmonically ...

, as is the one between ''B-flat'' and ''B'' (frequency 493 Hz). Accordingly, the frequency ratios agree:
:$\backslash frac\; \backslash approx\; \backslash frac\; \backslash approx\; 1.059\; \backslash approx\; \backslash sqrt;\; href="/html/ALL/s/2.html"\; ;"title="2">2$
Therefore, logarithms can be used to describe the intervals: an interval is measured in semitones by taking the logarithm of the frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in Hertz (unit), hertz (H ...

ratio, while the logarithm of the frequency ratio expresses the interval in cents, hundredths of a semitone. The latter is used for finer encoding, as it is needed for non-equal temperaments.
Number theory

Natural logarithm
The natural logarithm of a number is its logarithm to the base (exponentiation), base of the mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approximately equal to . The natur ...

s are closely linked to counting prime numbers (2, 3, 5, 7, 11, ...), an important topic in number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative intege ...

. For any integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...

, the quantity of prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only wa ...

s less than or equal to is denoted . The prime number theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic analysis, asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by pr ...

asserts that is approximately given by
:$\backslash frac,$
in the sense that the ratio of and that fraction approaches 1 when tends to infinity. As a consequence, the probability that a randomly chosen number between 1 and is prime is inversely proportional to the number of decimal digits of . A far better estimate of is given by the offset logarithmic integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

function , defined by
:$\backslash mathrm(x)\; =\; \backslash int\_2^x\; \backslash frac1\; \backslash ,dt.$
The Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its Root of a function, zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important List ...

, one of the oldest open mathematical conjecture
In mathematics, a conjecture is a Consequent, conclusion or a proposition that is proffered on a tentative basis without Formal proof, proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conje ...

s, can be stated in terms of comparing and . The Erdős–Kac theorem describing the number of distinct prime factor
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

s also involves the factorial
In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times ...

, , is given by
:$\backslash ln\; (n!)\; =\; \backslash ln\; (1)\; +\; \backslash ln\; (2)\; +\; \backslash cdots\; +\; \backslash ln\; (n).$
This can be used to obtain Stirling's formula, an approximation of for large .
Generalizations

Complex logarithm

All thecomplex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...

s that solve the equation
:$e^a=z$
are called ''complex logarithms'' of , when is (considered as) a complex number. A complex number is commonly represented as , where and are real numbers and is an imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimen ...

, the square of which is −1. Such a number can be visualized by a point in the complex plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, ...

, as shown at the right. The polar form
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...

encodes a non-zero complex number by its absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign (mathematics), sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative number, negative (in which cas ...

, that is, the (positive, real) distance to the origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...

, and an angle between the real () axis'' '' and the line passing through both the origin and . This angle is called the argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called a conclusion. Arguments can be studied from three main perspectives: the logical, the dialect ...

of .
The absolute value of is given by
:$\backslash textstyle\; r=\backslash sqrt.$
Using the geometrical interpretation of sine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...

and cosine and their periodicity in , any complex number may be denoted as
:$z\; =\; x\; +\; iy\; =\; r\; (\backslash cos\; \backslash varphi\; +\; i\; \backslash sin\; \backslash varphi\; )=\; r\; (\backslash cos\; (\backslash varphi\; +\; 2k\backslash pi)\; +\; i\; \backslash sin\; (\backslash varphi\; +\; 2k\backslash pi)),$
for any integer number . Evidently the argument of is not uniquely specified: both and are valid arguments of for all integers , because adding radians
The radian, denoted by the symbol rad, is the unit of angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called t ...

or ''k''⋅360° to corresponds to "winding" around the origin counter-clock-wise by turns. The resulting complex number is always , as illustrated at the right for . One may select exactly one of the possible arguments of as the so-called ''principal argument'', denoted , with a capital , by requiring to belong to one, conveniently selected turn, e.g. or . These regions, where the argument of is uniquely determined are called ''branches'' of the argument function.
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematics, mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex number, complex exponential function. Euler ...

connects the trigonometric functions
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

sine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...

and cosine to the complex exponential
The exponential function is a mathematical Function (mathematics), function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponentiation, exponent). Unless otherwise specified, the term generally refers to the positiv ...

:
:$e^\; =\; \backslash cos\; \backslash varphi\; +\; i\backslash sin\; \backslash varphi\; .$
Using this formula, and again the periodicity, the following identities hold:
:$\backslash beginz\&\; =\; \&\; r\; \backslash left\; (\backslash cos\; \backslash varphi\; +\; i\; \backslash sin\; \backslash varphi\backslash right)\; \backslash \backslash \; \&\; =\; \&\; r\; \backslash left\; (\backslash cos(\backslash varphi\; +\; 2k\backslash pi)\; +\; i\; \backslash sin(\backslash varphi\; +\; 2k\backslash pi)\backslash right)\; \backslash \backslash \; \&\; =\; \&\; r\; e^\; \backslash \backslash \; \&\; =\; \&\; e^\; e^\; \backslash \backslash \; \&\; =\; \&\; e^\; =\; e^,\; \backslash end$
where is the unique real natural logarithm, denote the complex logarithms of , and is an arbitrary integer. Therefore, the complex logarithms of , which are all those complex values for which the power of equals , are the infinitely many values
:$a\_k\; =\; \backslash ln\; (r)\; +\; i\; (\; \backslash varphi\; +\; 2\; k\; \backslash pi\; ),\backslash quad$ for arbitrary integers .
Taking such that is within the defined interval for the principal arguments, then is called the ''principal value'' of the logarithm, denoted , again with a capital . The principal argument of any positive real number is 0; hence is a real number and equals the real (natural) logarithm. However, the above formulas for logarithms of products and powers do ''not'' generalize to the principal value of the complex logarithm.
The illustration at the right depicts , confining the arguments of to the interval . This way the corresponding branch of the complex logarithm has discontinuities all along the negative real axis, which can be seen in the jump in the hue there. This discontinuity arises from jumping to the other boundary in the same branch, when crossing a boundary, i.e. not changing to the corresponding -value of the continuously neighboring branch. Such a locus is called a branch cut
In the mathematics, mathematical field of complex analysis, a branch point of a multivalued function, multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-val ...

. Dropping the range restrictions on the argument makes the relations "argument of ", and consequently the "logarithm of ", multi-valued function
In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function (mathematics), function, but may associate several values to each input. More precisely, a multivalued funct ...

s.
Inverses of other exponential functions

Exponentiation occurs in many areas of mathematics and its inverse function is often referred to as the logarithm. For example, thelogarithm of a matrix
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...

is the (multi-valued) inverse function of the matrix exponential
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...

. Another example is the ''p''-adic logarithm, the inverse function of the ''p''-adic exponential. Both are defined via Taylor series analogous to the real case. In the context of differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra ...

, the exponential map maps the tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of ''tangent plane (geometry), tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics ...

at a point of a manifold
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...

to a neighborhood
A neighbourhood (British English
British English (BrE, en-GB, or BE) is, according to Oxford Dictionaries, " English as used in Great Britain, as distinct from that used elsewhere". More narrowly, it can refer specifically to the En ...

of that point. Its inverse is also called the logarithmic (or log) map.
In the context of finite group
Finite is the opposite of Infinity, infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected ...

s exponentiation is given by repeatedly multiplying one group element with itself. The discrete logarithm
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...

is the integer ' solving the equation
:$b^n\; =\; x,$
where is an element of the group. Carrying out the exponentiation can be done efficiently, but the discrete logarithm is believed to be very hard to calculate in some groups. This asymmetry has important applications in public key cryptography
Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. Key pairs are generated with cryptographic alg ...

, such as for example in the Diffie–Hellman key exchange
Diffie–Hellman key exchangeSynonyms of Diffie–Hellman key exchange include:
* Diffie–Hellman–Merkle key exchange
* Diffie–Hellman key agreement
* Diffie–Hellman key establishment
* Diffie–Hellman key negotiation
* Exponential key exc ...

, a routine that allows secure exchanges of cryptographic
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logy, -logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of ...

keys over unsecured information channels. Zech's logarithm is related to the discrete logarithm in the multiplicative group of non-zero elements of a finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...

.
Further logarithm-like inverse functions include the ''double logarithm'' , the '' super- or hyper-4-logarithm'' (a slight variation of which is called iterated logarithm in computer science), the Lambert W function
In mathematics, the Lambert function, also called the omega function or product logarithm, is a multivalued function, namely the Branch point, branches of the converse relation of the function , where is any complex number and is the expone ...

, and the logit
In statistics, the logit ( ) function is the quantile function associated with the standard logistic distribution. It has many uses in data analysis and machine learning, especially in Data transformation (statistics), data transformations.
Ma ...

. They are the inverse functions of the double exponential function
A double exponential function is a Constant (mathematics), constant raised to the power of an Exponentiation, exponential function. The general formula is f(x) = a^=a^ (where ''a''>1 and ''b''>1), which grows much more quickly than an exponentia ...

, tetration
In mathematics, tetration (or hyper-4) is an operation (mathematics), operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though \uparrow \uparrow and the left-exponent ''xb'' are common.
Und ...

, of , and of the logistic function
A logistic function or logistic curve is a common S-shaped curve (sigmoid function, sigmoid curve) with equation
f(x) = \frac,
where
For values of x in the domain of real numbers from -\infty to +\infty, the S-curve shown on the right is ...

, respectively.
Related concepts

From the perspective ofgroup theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

, the identity expresses a group isomorphism
In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two ...

between positive reals under multiplication and reals under addition. Logarithmic functions are the only continuous isomorphisms between these groups. By means of that isomorphism, the Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.
This Measure (mathematics), measure was introduced by Alfré ...

(Lebesgue measure
In Measure (mathematics), measure theory, a branch of mathematics, the Lebesgue measure, named after france, French mathematician Henri Lebesgue, is the standard way of assigning a measure (mathematics), measure to subsets of ''n''-dimensional Eucl ...

) on the reals corresponds to the Haar measure on the positive reals. The non-negative reals not only have a multiplication, but also have addition, and form a semiring
In abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), ...

, called the probability semiring
In abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), ...

; this is in fact a semifield
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...

. The logarithm then takes multiplication to addition (log multiplication), and takes addition to log addition ( LogSumExp), giving an isomorphism
In mathematics, an isomorphism is a structure-preserving Map (mathematics), mapping between two Mathematical structure, structures of the same type that can be reversed by an inverse function, inverse mapping. Two mathematical structures are iso ...

of semirings between the probability semiring and the log semiring
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...

.
Logarithmic one-forms appear in complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

and algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. ...

as differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...

s with logarithmic poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavs, West Slavic nation and ethnic group, who share a common History of Poland, history, Culture of Poland, culture, the Polish language and ...

.
The polylogarithm
In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function of order and argument . Only for special values of does the polylogarithm reduce to an elementary function such as the natura ...

is the function defined by
:$\backslash operatorname\_s(z)\; =\; \backslash sum\_^\backslash infty\; .$
It is related to the Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter
The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BCE. It is derived from the earlier Phoenician al ...

.
See also

*Decimal exponent
In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs (mathematician), Henry Briggs, an English ...

(dex)
* Exponential function
The exponential function is a mathematical Function (mathematics), function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponentiation, exponent). Unless otherwise specified, the term generally refers to the positiv ...

* Index of logarithm articles
Notes

References

External links

* * * *Khan Academy: Logarithms, free online micro lectures

* * * * {{Authority control Elementary special functions Scottish inventions Additive functions