TheInfoList In
mathematics Mathematics (from Greek: ) includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (analysis). It has no generally accepted definition. Mathematicians seek and use patterns to formulate ...
, the logarithm is the
inverse function In mathematics, an inverse function (or anti-function) is a function that "reverses" another function: if the function applied to an input gives a result of , then applying its inverse function to gives the result , i.e., if and only if . T ...
to
exponentiation Exponentiation is a mathematical operation, written as , involving two numbers, the ''base'' and the ''exponent'' or ''power'' , and pronounced as " raised to the power of ". When is a positive integer, exponentiation corresponds to repeated ...
. That means the logarithm of a given number  is the
exponent Exponentiation is a mathematical operation, written as , involving two numbers, the ''base'' and the ''exponent'' or ''power'' , and pronounced as " raised to the power of ". When is a positive integer, exponentiation corresponds to repeated ...
to which another fixed number, the ''
base Base or BASE may refer to: Brands and enterprises *Base (mobile telephony provider), a Belgian mobile telecommunications operator *Base CRM, an enterprise software company founded in 2009 with offices in Mountain View and Kraków, Poland *Base De ...
'' , must be raised, to produce that number . In the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication; e.g., since , the "logarithm base " 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 limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and ...
. More generally, exponentiation allows any positive
real number Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ... as base to be raised to any real power, always producing a positive result, so for any two positive real numbers and , where is not equal to , is always a unique real number . More explicitly, the defining relation between exponentiation and logarithm is: :$\log_b\left(x\right) = y \$ exactly if $\ b^y = x\$ and $\ x > 0$ and $\ b > 0$ and $\ b \ne 1$. For example, , as . The logarithm base (that is ) 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, an English mathematician who pioneered its ... and is commonly used in science and engineering. The natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , where is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if the ... has the number (that is ) as its base; its use is widespread in mathematics and physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its motion and behavior through space and time, and the related ent ... , because of its simpler integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with diffe ... and derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For ... . The binary logarithm In mathematics, the binary logarithm () is the power to which the number must be raised to obtain the value . That is, for any real number , :x=\log_2 n \quad\Longleftrightarrow\quad 2^x=n. For example, the binary logarithm of is , the bin ... uses base (that is ) and is frequently used in computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of algorithmic processes, comp ... . Logarithms are examples of concave function In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap or upper convex. Definition A real-valued function f on an interva ... s. 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 Latinized name was Ioanne ... 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 of the fact—important in its own right—that 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 * Produc ... is the sum of the logarithms of the factors: :$\log_b\left(xy\right) = \log_b x + \log_b y, \,$ provided that , and are all positive and . The slide rule#REDIRECT Slide rule {{R from other capitalisation ... , 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 made important and influential discoveries in many branches of mathematics, such as infinitesimal c ... , who connected them to the exponential function In mathematics, an exponential function is a function of the form where is a positive real number, and the argument occurs as an exponent. For real numbers and , a function of the form f(x)=ab^ is also an exponential function, since it can ... in the 18th century, and who also introduced the letter as the base of natural logarithms. Logarithmic scaleA 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 sc ... s reduce wide-ranging quantities to tiny scopes. For example, the decibel The decibel (symbol: dB) is a relative unit of measurement corresponding to one tenth of a bel (B). It is used to express the ratio of one value of a power or root-power quantity to another, on a logarithmic scale. A logarithmic quantity in deci ... (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 donating a proton (hydrogen ion H+) (a Brønsted–Lowry acid), or, alternatively, capable of forming a covalent bond with an electron pair (a Lewis acid). The first category of acids are the proton do ... ity of an aqueous solution An aqueous solution is a 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), in water would be repr ... . 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 term ''formula'' in science refers to the general construct of a relationship between given ... e, and in measurements of the complexity of algorithms and of geometric objects called fractal In mathematics, a fractal is a subset of Euclidean space with a fractal dimension that strictly exceeds its topological dimension. Fractals appear the same at different scales, as illustrated in successive magnifications of the Mandelbrot set. ... s. They help to describe frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also referred to as temporal frequency, which emphasizes the contrast to spatial frequency and angular frequency. Frequency is measured in hertz (Hz) which is e ... 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 ways o ... s or approximating factorial In mathematics, the factorial of a non-negative integer , denoted by , is the product of all positive integers less than or equal to : :n! = n \cdot (n-1) \cdot (n-2) \cdot (n-3) \cdot \cdots \cdot 3 \cdot 2 \cdot 1 \,. For example, :5! = ... s, inform some models in psychophysics Psychophysics quantitatively investigates the relationship between physical stimuli and the sensations and perceptions they produce. Psychophysics has been described as "the scientific study of the relation between stimulus and sensation" or, more ... , and can aid in forensic accounting Forensic accounting, forensic accountancy or financial forensics is the specialty practice area of accounting that investigates whether firms engage in financial reporting misconduct. Forensic accountants apply a range of skills and methods to d ... . In the same way as the logarithm reverses exponentiation Exponentiation is a mathematical operation, written as , involving two numbers, the ''base'' and the ''exponent'' or ''power'' , and pronounced as " raised to the power of ". When is a positive integer, exponentiation corresponds to repeated ... , the complex logarithm of the color is used to show the ''arg'' (polar coordinate angle) of the complex logarithm. The saturation and value (intensity and brightness) of the color is used to show the ''modulus'' of the complex logarithm. In the branch of mathematics kn ... is the inverse function In mathematics, an inverse function (or anti-function) is a function that "reverses" another function: if the function applied to an input gives a result of , then applying its inverse function to gives the result , i.e., if and only if . T ... of the exponential function, whether applied to real number Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ...
s or
complex number In mathematics, a complex number is a number that can be expressed in the form , where and are real numbers, and is a symbol called the imaginary unit, and satisfying the equation . Because no "real" number satisfies this equation, was called ...
s. The modular
discrete logarithm In mathematics, for given real numbers ''a'' and ''b'', the logarithm log''b'' ''a'' is a number ''x'' such that . Analogously, in any group ''G'', powers ''b'k'' can be defined for all integers ''k'', and the discrete logarithm log''b''& ...
is another variant; it has uses in
public-key cryptography 250px, In this example the message is digitally signed, but not encrypted. 1) Alice signs a message with her private key. 2) Bob can verify that Alice sent the message and that the message has not been modified. Public-key cryptography, or as ...
.

# Motivation and definition Addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or ''sum'' of those ...
,
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, ...
, and
exponentiation Exponentiation is a mathematical operation, written as , involving two numbers, the ''base'' and the ''exponent'' or ''power'' , and pronounced as " raised to the power of ". When is a positive integer, exponentiation corresponds to repeated ...
are three of the most fundamental arithmetic operations. Addition, the simplest of these, is undone by subtraction: when you add to to get , to reverse this operation you need to ''subtract'' from . Multiplication, the next-simplest operation, is undone by
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting o ...
: if you multiply by to get , you then can divide by to return to the original expression . Logarithms also undo a fundamental arithmetic operation, exponentiation. Exponentiation is when you raise a number to a certain power. For example, raising to the power equals : : $2^3 = 2 \times 2 \times 2 = 8$ The general case is when you raise a number to the power of to get : : $b^y=x$ The number is referred to as the base of this expression. The base is the number that is raised to a particular power—in the above example, the base of the expression $2^3=8$ is . It is easy to make the base the subject of the expression: all you have to do is take the root of both sides. This gives: :

## Exponentiation

This subsection contains a short overview of the exponentiation operation, which is fundamental to understanding logarithms. Raising to the power, where is a
natural number In mathematics, the natural numbers are those 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"). In common mathematical terminology, words colloquially used ...
, is done by multiplying factors equal to . The power of is written , so that : $b^n = \underbrace_$ Exponentiation may be extended to , where is a positive number and the ''exponent'' is any
real number Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R\$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ...
. For example, is the
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another poly ...
of , that is, . Raising ' to the power 1/2 is the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . Eve ...
of '. More generally, raising ' to a
rational Rationality is the quality or state of being rational – that is, being based on or agreeable to reason. Rationality implies the conformity of one's beliefs with one's reasons to believe, and of one's actions with one's reasons for action. "Ra ...
power , where and are integers, is given by : the -th root of $b^p\!\!$. Finally, any
irrational number In mathematics, the irrational numbers are all the real numbers which are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, ...
(a real number which is not rational) ' can be approximated to arbitrary precision by rational numbers. This can be used to compute the '-th power of ': for example $\sqrt 2 \approx 1.414 ...$ and $b^$ is increasingly well approximated by $b^1, b^, b^, b^, ...$. A more detailed explanation, as well as the formula is contained in the article on
exponentiation Exponentiation is a mathematical operation, written as , involving two numbers, the ''base'' and the ''exponent'' or ''power'' , and pronounced as " raised to the power of ". When is a positive integer, exponentiation corresponds to repeated ...
.

## Definition

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 solution to the equation : $b^y = x.$ The logarithm is denoted "" (pronounced as "the logarithm of to base " or "the logarithm of " or (most commonly) "the log, base , of "). In the equation , the value is the answer to the question "To what power must be raised, in order to yield ?".

## Examples

* , since . * Logarithms can also be negative: $\quad \log_2 \! \frac = -1 \quad$ since $\quad 2^ = \frac = \frac.$ * 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: :$\log_b x = \frac.\,$ Starting from the defining identity : $x = b^$ we can apply to both sides of this equation, to get : $\log_k x = \log_k \left\left(b^\right\right) = \log_b x \cdot \log_k b$. Solving for $\log_b x$ yields: : $\log_b x = \frac$, showing the conversion factor from given $\log_k$-values to their corresponding $\log_b$-values to be $\left(\log_k b\right)^.$ Typical
scientific calculators Casio fx-77, a solar-powered digital calculator from the 1980s using a single-line LCD A digital calculator is a type of electronic calculator, usually but not always handheld, designed to calculate problems in science, engineering, and mathemat ...
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: :$\log_b x = \frac = \frac. \,$ Given a number and its logarithm to an unknown base , the base is given by: :$b = x^\frac,$ which can be seen from taking the defining equation $x = b^ = b^y$ to the power of $\; \tfrac.$

# Particular bases Among all choices for the base, three are particularly common. These are , (the
irrational 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. The ...
mathematical constant ≈ 2.71828), and (the
binary logarithm In mathematics, the binary logarithm () is the power to which the number must be raised to obtain the value . That is, for any real number , :x=\log_2 n \quad\Longleftrightarrow\quad 2^x=n. For example, the binary logarithm of is , the bin ...
). In
mathematical analysis#REDIRECT Mathematical analysis#REDIRECT Mathematical analysis {{R from other capitalisation ...
{{R from other capitalisation ...
, 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 occasionally called denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hi ...
number system: :$\log_\left(10 x\right) = \log_ 10 + \log_ x = 1 + \log_ x.\$ Thus, is related to the number of
decimal digit , in order of value. A numerical digit is a single symbol used alone (such as "2") or in combinations (such as "25"), to represent numbers in a positional numeral system. The name "digit" comes from the fact that the ten digits (Latin ''digiti'' me ...
s of a positive integer : the number of digits is the smallest
integer An integer (from the Latin ''integer'' meaning "whole") is colloquially defined as a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and  are not. The set of intege ...
strictly bigger than log10 ''x''. 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, storage, and communication of information. The field was fundamentally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. T ...
, corresponding to the use of nats or
bit The bit is a basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represented as either , ...
s as the fundamental units of information, respectively. Binary logarithms are also used in
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of algorithmic processes, comp ...
, 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 animated examples)''. More restrictive definitions require that this ...
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 "rudiments", that are needed to understand music notation (key ...
, where a pitch ratio of two (the
octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been referred to ...
) is ubiquitous and the
cent Cent may refer to: Currency * Cent (currency), a one-hundredth subdivision of several units of currency * Penny (Canadian coin), a Canadian coin removed from circulation in 2013 * 1 cent (Dutch coin), a Dutch coin minted between 1941 and 1944 * 1 ...
is the binary logarithm (scaled by 1200) of the ratio between two adjacent equally-tempered pitches in European
classical music Classical music is art music produced or rooted in the traditions of Western culture, including both liturgical (religious) and secular music. Historically, the term 'classical music' refers specifically to the musical period from 1750 to 1820 (t ...
; and in
photography Photography is the 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 is employed in ...
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 have the same EV (for any fixed scene luminance). Exposure value is als ...
s. 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-setting body composed of representatives from various national standards organizations. Founded on 23 February 1947, the organization promotes worldwide p ...
(
ISO 80000-2 ISO 80000 or IEC 80000 is an international standard introducing the International System of Quantities (ISQ). It was developed and promulgated jointly by the International Organization for Standardization (ISO) and the International Electrotechnic ...
). 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 Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriented ...
that extended the realm of analysis beyond the scope of algebraic methods. The method of logarithms was publicly propounded 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 Latinized name was Ioanne ... 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 using formulas from trigonometry. For the 25 years preceding the i ...
or the use of tables of progressions, extensively developed by
Jost Bürgi Jost Bürgi (also ''Joost, Jobst''; Latinized surname ''Burgius'' or ''Byrgius''; 28 February 1552 – 31 January 1632), active primarily at the courts in Kassel and Prague, was a Swiss clockmaker, a maker of astronomical instruments and a mathe ...
around 1600. Napier coined the term for logarithm in Middle Latin, "logarithmorum," 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, an English mathematician who pioneered its ...
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 (; grc, ; ; ) was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Considered to be ... 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 using formulas from trigonometry. For the 25 years preceding the i ...
. Invention of the
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriented ...
now known as the
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , where is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if the ...
began as an attempt to perform a quadrature of a rectangular
hyperbola 300px, Hyperbola (red): features In mathematics, a hyperbola () (adjective form hyperbolic, ) (plural ''hyperbolas'', or ''hyperbolae'' ()) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it ... by
Grégoire de Saint-Vincent Grégoire de Saint-Vincent - in latin : Gregorius a Sancto Vincentio, in dutch : Gregorius van St-Vincent - (8 September 1584 Bruges – 5 June 1667 Ghent) was a Flemish Jesuit and mathematician. He is remembered for his work on quadrature of the ...
, a Belgian Jesuit residing in Prague. Archimedes had written ''
The Quadrature of the Parabola ''The Quadrature of the Parabola'' ( el, Τετραγωνισμὸς παραβολῆς) is a treatise on geometry, written by Archimedes in the 3rd century BC. Written as a letter to his friend Dositheus, the work presents 24 propositions regardi ...
'' 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 exam ...
in its
argument In logic and philosophy, an argument is a series of statements (in a natural language), called the premises or premisses (both spellings are acceptable), intended to determine the degree of truth of another statement, the conclusion. The logical ...
and an
arithmetic progression An Arithmetic progression (AP) 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 diffe ...
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 using formulas from trigonometry. For the 25 years preceding the i ...
, leading to the term "hyperbolic logarithm", a synonym for natural logarithm. Soon the new function was appreciated by
Christiaan Huygens Christiaan Huygens ( , also , ; la, Hugenius; 14 April 1629 – 8 July 1695), also spelled Huyghens, was a Dutch mathematician, physicist, astronomer and inventor, who is widely regarded as one of the greatest scientists of all time and a majo ... , and James Gregory. The notation Log y was adopted by
Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the "1666–1676" section. (; or ; – 14 November 1716) was a prominent German polymath and one of the most important logicians, mathematicians and natural philoso ... in 1675, and the next year he connected it to the
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with diffe ...
$\int \frac .$ Before Euler developed his modern conception of complex natural logarithms,
Roger Cotes Roger Cotes FRS (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 ''Principia'', before publication. He also invented the quadrature ...
had a nearly equivalent result when he showed in 1714 that :$\log\left(\cos \theta + i\sin \theta\right) = i\theta$

# Logarithm tables, slide rules, and historical applications

'' explanation of logarithms By simplifying difficult calculations before calculators and computers became available, logarithms contributed to the advance of science, especially
astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain t ...
. They were critical to advances in
surveying Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial or three-dimensional positions of points and the distances and angles between them. A land surveying professional is called a land surveyo ...
,
celestial navigation Celestial navigation, also known as astronavigation, is the ancient and modern practice of position fixing that enables a navigator to transition through a space without having to rely on estimated calculations, or dead reckoning, to know their p ...
, 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 a ... 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 log''b'' ''x'', it has been called the antilogarithm.

## 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 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, an English mathematician who pioneered its ...
s of all integers in the range 1–1000, 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 In mathematics and computer science, the floor function is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted \operatorname(x) or \lfloor x\rfloor. Similarly, the ceilin ...
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 of or \lfloor x\rfloor, its fractional part can b ...
, known as the characteristic and
mantissa Mantissa () may refer to: * Mantissa (logarithm), the fractional part of the common (base-10) logarithm * Mantissa (floating point number), the significant digits of a floating-point number or a number in scientific notation, also called the ''sign ...
. 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 :$\log_3542 = \log_\left(1000 \cdot 3.542\right) = 3 + \log_3.542 \approx 3 + \log_3.54 \,$ Greater accuracy can be obtained by
interpolation In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing new data points within the range of a discrete set of known data points. In engineering and science, one often has a number of data ...
: :$\log_3542 \approx 3 + \log_3.54 + 0.2 \left(\log_3.55-\log_3.54\right)\,$ The value of can be determined by reverse look up in the same table, since the logarithm is a
monotonic function Figure 3. A function that is not monotonic In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized ...
.

## 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: :$c d = 10^ \, 10^ = 10^ \,$ and :$\frac c d = c d^ = 10^. \,$ 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 as
prosthaphaeresis Prosthaphaeresis (from the Greek ''προσθαφαίρεσις'') was an algorithm used in the late 16th century and early 17th century for approximate multiplication and division using formulas from trigonometry. For the 25 years preceding the i ...
, which relies on
trigonometric identities In mathematics, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving ce ...
. Calculations of powers and
roots A root is the part of a plant that most often lies below the surface of the soil but can also be aerial or aerating, that is, growing up above the ground or especially above water. Root or roots may also refer to: Art, entertainment, and media * ...
are reduced to multiplications or divisions and look-ups by :$c^d = \left\left(10^\right\right)^d = 10^ \,$ and : 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 al ...
s.

## Slide rules

Another critical application was the
slide rule#REDIRECT Slide rule {{R from other capitalisation ...
, 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) was an English mathematician and Anglican clergyman. After John Napier invented logarithms and Edmund Gunter created the logarithmic scales (lines, or rules) upon which slide rules are based, O ...
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: Image:Slide rule example2 with labels.svg, center, 550px, Schematic depiction of a slide rule. Starting from 2 on the lower scale, add the distance to 3 on the upper scale to reach the product 6. The slide rule works because it is marked such that the distance from 1 to is proportional to the logarithm of ., alt=A slide rule: two rectangles with logarithmically ticked axes, arrangement to add the distance from 1 to 2 to the distance from 1 to 3, indicating the product 6. 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 Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriented ...
''. 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: $f\left(x\right) = b^x. \,$

## Logarithmic function

To justify the definition of logarithms, it is necessary to show that the equation :$b^x = y \,$ has a solution and that this solution is unique, provided that is positive and that is positive and unequal to 1. A proof of that fact requires the
intermediate value theorem In mathematical analysis, the intermediate value theorem states that if ''f'' is a continuous function whose domain contains the interval 'a'', ''b'' then it takes on any given value between ''f''(''a'') and ''f''(''b'') at some point within ...
from elementary
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic ...
., section III.3 This theorem states that a
continuous function In mathematics, a continuous function is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to suff ...
that produces two values ' and ' also produces any value that lies between ' and '. A function is ''continuous'' if it does not "jump", that is, if its graph can be drawn without lifting the pen. This property can be shown to hold for the function . Because ' takes arbitrarily large and arbitrarily small positive values, any number lies between and for suitable and . Hence, the intermediate value theorem ensures that the equation has a solution. Moreover, there is only one solution to this equation, because the function ''f'' is
strictly increasing Figure 3. A function that is not monotonic In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized ...
(for ), or strictly decreasing (for ). The unique solution is the logarithm of to base , . The function that assigns to its logarithm is called ''logarithm function'' or ''logarithmic function'' (or just ''logarithm''). The function is essentially characterized by the product formula :$\log_b\left(xy\right) = \log_b x + \log_b y.$ More precisely, the logarithm to any base is the only
increasing function Figure 3. A function that is not monotonic In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized ...
''f'' from the positive reals to the reals satisfying and :$f\left(xy\right)=f\left(x\right)+f\left(y\right).$

## Inverse function The formula for the logarithm of a power says in particular that for any number , :$\log_b \left \left(b^x \right\right) = x \log_b b = x.$ In prose, taking the power of and then the logarithm gives back . Conversely, given a positive number , the formula :$b^ = y$ says that first taking the logarithm and then exponentiating gives back . Thus, the two possible ways of combining (or composing) logarithms and exponentiation give back the original number. Therefore, the logarithm to base is the ''
inverse function In mathematics, an inverse function (or anti-function) is a function that "reverses" another function: if the function applied to an input gives a result of , then applying its inverse function to gives the result , i.e., if and only if . T ...
'' of . Inverse functions are closely related to the original functions. 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 ''f'' 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 an
increasing function Figure 3. A function that is not monotonic In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized ...
. 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 and
differentiable function In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each ...
, so is . Roughly, a continuous function is differentiable if its graph has no sharp "corners". Moreover, as the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For ...
of evaluates to by the properties of the
exponential function In mathematics, an exponential function is a function of the form where is a positive real number, and the argument occurs as an exponent. For real numbers and , a function of the form f(x)=ab^ is also an exponential function, since it can ...
, the
chain rule#REDIRECT chain rule#REDIRECT chain rule {{Redirect category shell, 1= {{R from other capitalisation ...
{{Redirect category shell, 1= {{R from other capitalisation ...
implies that the derivative of is given by : $\frac \log_b x = \frac.$ That is, the
slope In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is used ...
of the
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More pre ... touching the graph of the logarithm at the point equals . The derivative of ln is 1/''x''; this implies that ln is the unique
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
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 generalised functional argument is :$\frac \ln f\left(x\right) = \frac.$ The quotient at the right hand side is called the
logarithmic derivative In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function ''f'' is defined by the formula : \frac where f' is the derivative of ''f''. Intuitively, this is the infinitesimal relative change in ''f'' ...
of ''f''. Computing by means of the derivative of is known as logarithmic differentiation. The antiderivative of the
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , where is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if the ...
is: : $\int \ln\left(x\right) \,dx = x \ln\left(x\right) - x + C.$ List of integrals of logarithmic functions, 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 The
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , where is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if the ...
of can be defined as the definite integral: :$\ln t = \int_1^t \frac \, dx.$ 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 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: :$\ln\left(tu\right) = \int_1^ \frac \, dx \ \stackrel = \int_1^ \frac \, dx + \int_t^ \frac \, dx \ \stackrel = \ln\left(t\right) + \int_1^u \frac \, dw = \ln\left(t\right) + \ln\left(u\right).$ 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 ''t'' 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 ''t'' to ''tu'' is the same as the integral from 1 to ''u''. This justifies the equality (2) with a more geometric proof. The power formula may be derived in a similar way: :$\ln\left(t^r\right) = \int_1^ \fracdx = \int_1^t \frac \left\left(rw^ \, dw\right\right) = r \int_1^t \frac \, dw = r \ln\left(t\right).$ The second equality uses a change of variables (integration by substitution), . The sum over the reciprocals of natural numbers, :$1 + \frac 1 2 + \frac 1 3 + \cdots + \frac 1 n = \sum_^n \frac,$ is called the harmonic series (mathematics), harmonic series. It is closely tied to the
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , where is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if the ...
: as ''n'' tends to infinity, the difference, :$\sum_^n \frac - \ln\left(n\right),$ limit of a sequence, converges (i.e., gets arbitrarily close) to a number known as the Euler–Mascheroni constant . This relation aids in analyzing the performance of algorithms such as quicksort.

## Transcendence of the logarithm

Real numbers that are not Algebraic number, algebraic are called transcendental number, transcendental; for example, Pi, and ''e (mathematical constant), e'' are such numbers, but $\sqrt$ is not. Almost all real numbers are transcendental. The logarithm is an example of a transcendental function. 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 using power series or the arithmetic–geometric mean, or be retrieved from a precalculated logarithm table that provides a fixed precision. Newton's method, 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-like methods can be used to compute logarithms by using only the operations of addition and Arithmetic shift, bit shifts. Moreover, the Binary logarithm#Algorithm, binary logarithm algorithm calculates recursion, recursively, based on repeated squarings of , taking advantage of the relation :$\log_2\left\left(x^2\right\right) = 2 \log_2 , x, .$

## Power series

;Taylor series For any real number that satisfies , the following formula holds: :$\begin\ln \left(z\right) &= \frac - \frac + \frac - \frac + \cdots \\ &= \sum_^\infty \left(-1\right)^\frac \end$ This is a shorthand for saying that can be approximated to a more and more accurate value by the following expressions: :$\begin \left(z-1\right) & & \\ \left(z-1\right) & - & \frac & \\ \left(z-1\right) & - & \frac & + & \frac \\ \vdots & \end$ For example, with the third approximation yields 0.4167, which is about 0.011 greater than . This series (mathematics), series approximates with arbitrary precision, provided the number of summands is large enough. In elementary calculus, is therefore the limit (mathematics), limit of this series. It is the Taylor series of the
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , where is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if the ...
at . The Taylor series of provides a particularly useful approximation to when is small, , since then :$\ln \left(1+z\right) = z - \frac +\frac\cdots \approx z.$ For example, with the first-order approximation gives , which is less than 5% off the correct value 0.0953. ;More efficient series Another series is based on the area hyperbolic tangent function: :$\ln \left(z\right) = 2\cdot\operatorname\,\frac = 2 \left \left( \frac + \frac^3 + \frac^5 + \cdots \right \right),$ for any real number . Using sigma notation, this is also written as :$\ln \left(z\right) = 2\sum_^\infty\frac\left\left(\frac\right\right)^.$ This series can be derived from the above Taylor series. It converges more quickly 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 :$A = \frac z, \,$ the logarithm of is: :$\ln \left(z\right)=y+\ln \left(A\right). \,$ The better the initial approximation is, the closer is to 1, so its logarithm can be calculated efficiently. can be calculated using the exponential function, 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 $\textstyle z=\frac$ in the above series, it follows that: :$\ln \left(n+1\right) = \ln\left(n\right) + 2\sum_^\infty\frac\left\left(\frac\right\right)^.$ If the logarithm of a large integer is known, then this series yields a fast converging series for , with a rate of convergence of $\frac$.

## Arithmetic–geometric mean approximation

The arithmetic–geometric mean yields high precision approximations of the
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , where is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if the ...
. 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 ''p'' precise bits) by the following formula (due to Carl Friedrich Gauss): :$\ln \left(x\right) \approx \frac - m \ln \left(2\right).$ Here denotes the arithmetic–geometric mean of and . It is obtained by repeatedly calculating the average $\left(x+y\right)/2$ (arithmetic mean) and $\sqrt$ (geometric mean) 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 . ''m'' is chosen such that :$x \,2^m > 2^.\,$ to ensure the required precision. A larger ''m'' makes the calculation take more steps (the initial x and y 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 at Los Alamos National Laboratory working on the Manhattan Project, Richard Feynman developed a bit-processing algorithm that is similar to long division and was later used in the Connection Machine. The algorithm uses the fact that every real number $1 < x < 2$ is representable as a product of distinct factors of the form $1 + 2^$. The algorithm sequentially builds that product $P$: if $P \cdot \left(1 + 2^\right) < x$, then it changes $P$ to $P \cdot \left(1 + 2^\right)$. It then increases $k$ by one regardless. The algorithm stops when $k$ is large enough to give the desired accuracy. Because $\log\left(x\right)$ is the sum of the terms of the form $\log\left(1 + 2^\right)$ corresponding to those $k$ for which the factor $1 + 2^$ was included in the product $P$, $\log\left(x\right)$ may be computed by simple addition, using a table of $\log\left(1 + 2^\right)$ for all $k$. 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 of scale invariance. For example, each chamber of the shell of a nautilus is an approximate copy of the next one, scaled by a constant factor. This gives rise to a logarithmic spiral. Benford's law on the distribution of leading digits can also be explained by scale invariance. Logarithms are also linked to self-similarity. 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 scaleA 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 sc ... 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, the Fenske equation, or the Nernst equation.

## Logarithmic scale Scientific quantities are often expressed as logarithms of other quantities, using a ''logarithmic scale''. For example, the
decibel The decibel (symbol: dB) is a relative unit of measurement corresponding to one tenth of a bel (B). It is used to express the ratio of one value of a power or root-power quantity to another, on a logarithmic scale. A logarithmic quantity in deci ...
is a unit of measurement associated with logarithmic-scale level quantity, quantities. It is based on the common logarithm of ratios—10 times the common logarithm of a power (physics), power ratio or 20 times the common logarithm of a voltage ratio. It is used to quantify the loss of voltage levels in transmitting electrical signals, to describe power levels of sounds in acoustics, and the absorbance of light in the fields of spectrometer, spectrometry and optics. The signal-to-noise ratio describing the amount of unwanted noise (electronic), noise in relation to a (meaningful) signal (information theory), signal is also measured in decibels. In a similar vein, the peak signal-to-noise ratio is commonly used to assess the quality of sound and image compression 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 or the Richter magnitude scale. For example, a 5.0 earthquake releases 32 times and a 6.0 releases 1000 times the energy of a 4.0. Another logarithmic scale is apparent magnitude. It measures the brightness of stars logarithmically. Yet another example is pH in chemistry; pH is the negative of the common logarithm of the Activity (chemistry), activity of hydronium ions (the form hydrogen ions take in water). The activity of hydronium ions in neutral water is 10−7 molar concentration, 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 104 of the activity, that is, vinegar's hydronium ion activity is about . Semi-log plot, 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 In mathematics, an exponential function is a function of the form where is a positive real number, and the argument occurs as an exponent. For real numbers and , a function of the form f(x)=ab^ is also an exponential function, since it can ...
s of the form appear as straight lines with
slope In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is used ...
equal to the logarithm of . Log-log plot, 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 ''k''. This is applied in visualizing and analyzing power laws.

## 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 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 stimuli and the sensations and perceptions they produce. Psychophysics has been described as "the scientific study of the relation between stimulus and sensation" or, more ...
, the Weber–Fechner law proposes a logarithmic relationship between stimulus (psychology), stimulus and sensation (psychology), sensation 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.) 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 File:Benfords law illustrated by world's countries population.png, Distribution of first digits (in %, red bars) in the List of countries by population, population of the 237 countries of the world. Black dots indicate the distribution predicted by Benford's law., alt=A bar chart and a superimposed second chart. The two differ slightly, but both decrease in a similar fashion. Logarithms arise in probability theory: the law of large numbers dictates that, for a fair coin, as the number of coin-tosses increases to infinity, the observed proportion of heads binomial distribution, 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 distributions. When the logarithm of a random variable has a normal distribution, 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 of parametric statistical models. For such a model, the likelihood function depends on at least one parametric model, parameter 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 independence (probability), independent random variables. Benford's law describes the occurrence of digits in many data sets, 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 ''d'' (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.

## Computational complexity

Analysis of algorithms is a branch of
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of algorithmic processes, comp ...
that studies the time complexity, performance of algorithms (computer programs solving a certain problem)., pp. 1–2 Logarithms are valuable for describing algorithms that Divide and conquer algorithm, 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 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 ''N'' is the list's length. Similarly, the merge sort 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 big O notation, 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 Logarithmic growth, 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 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"). In common mathematical terminology, words colloquially used ...
''N'' can be represented in Binary numeral system, binary form in no more than
bit The bit is a basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represented as either , ...
s. In other words, the amount of memory (computing), memory needed to store ''N'' grows logarithmically with ''N''.

## Entropy and chaos Entropy is broadly a measure of the disorder of some system. In statistical thermodynamics, the entropy ''S'' of some physical system is defined as :$S = - k \sum_i p_i \ln\left(p_i\right).\,$ The sum is over all possible states ''i'' of the system in question, such as the positions of gas particles in a container. Moreover, is the probability that the state ''i'' is attained and ''k'' is the Boltzmann constant. Similarly, entropy (information theory), entropy in information theory measures the quantity of information. If a message recipient may expect any one of ''N'' 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. 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 chaos theory, chaotic in a Deterministic system, deterministic 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 the fractal dimension, dimension of
fractal In mathematics, a fractal is a subset of Euclidean space with a fractal dimension that strictly exceeds its topological dimension. Fractals appear the same at different scales, as illustrated in successive magnifications of the Mandelbrot set. ...
s. Fractals are geometric objects that are self-similarity, self-similar: 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 of this structure . Another logarithm-based notion of dimension is obtained by box-counting dimension, counting the number of boxes needed to cover the fractal in question.

## Music

Logarithms are related to musical tones and interval (music), intervals. In equal temperament, the frequency ratio depends only on the interval between two tones, not on the specific frequency, or pitch (music), pitch, of the individual tones. For example, the a (musical note), note ''A'' has a frequency of 440 Hertz, Hz and B♭ (musical note), ''B-flat'' has a frequency of 466 Hz. The interval between ''A'' and ''B-flat'' is a semitone, as is the one between ''B-flat'' and b (musical note), ''B'' (frequency 493 Hz). Accordingly, the frequency ratios agree: :$\frac \approx \frac \approx 1.059 \approx \sqrt\left[12\right]2.$ 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 referred to as temporal frequency, which emphasizes the contrast to spatial frequency and angular frequency. Frequency is measured in hertz (Hz) which is e ...
ratio, while the logarithm of the frequency ratio expresses the interval in cent (music), cents, hundredths of a semitone. The latter is used for finer encoding, as it is needed for non-equal temperaments.

## Number theory

Natural logarithms are closely linked to prime-counting function, counting prime numbers (2, 3, 5, 7, 11, ...), an important topic in number theory. For any
integer An integer (from the Latin ''integer'' meaning "whole") is colloquially defined as a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and  are not. The set of intege ...
, 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 ways o ...
s less than or equal to is denoted . The prime number theorem asserts that is approximately given by :$\frac,$ 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 proportionality (mathematics), proportional to the number of decimal digits of . A far better estimate of is given by the logarithmic integral function, offset logarithmic integral function , defined by :$\mathrm\left(x\right) = \int_2^x \frac1 \,dt.$ The Riemann hypothesis, one of the oldest open mathematical conjectures, can be stated in terms of comparing and . The Erdős–Kac theorem describing the number of distinct prime factors also involves the
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , where is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if the ...
. The logarithm of ''n''
factorial In mathematics, the factorial of a non-negative integer , denoted by , is the product of all positive integers less than or equal to : :n! = n \cdot (n-1) \cdot (n-2) \cdot (n-3) \cdot \cdots \cdot 3 \cdot 2 \cdot 1 \,. For example, :5! = ...
, , is given by :$\ln \left(n!\right) = \ln \left(1\right) + \ln \left(2\right) + \cdots + \ln \left(n\right). \,$ This can be used to obtain Stirling's formula, an approximation of for large ''n''.

# Generalizations

## Complex logarithm All the
complex number In mathematics, a complex number is a number that can be expressed in the form , where and are real numbers, and is a symbol called the imaginary unit, and satisfying the equation . Because no "real" number satisfies this equation, was called ...
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 square of which is −1. Such a number can be visualized by a point in the complex plane, as shown at the right. The polar form encodes a non-zero complex number by its absolute value, that is, the (positive, real) distance to the origin (mathematics), origin, and an angle between the real () axis ''Re'' and the line passing through both the origin and . This angle is called the Argument (complex analysis), argument of . The absolute value of is given by :$\textstyle r=\sqrt.$ Using the geometrical interpretation of $\sin$ and $\cos$ and their periodicity in $2\pi,$ any complex number may be denoted as :$z = x + iy = r \left(\cos \varphi + i \sin \varphi \right)= r \left(\cos \left(\varphi + 2k\pi\right) + i \sin \left(\varphi + 2k\pi\right)\right),$ for any integer number . Evidently the argument of is not uniquely specified: both and ' = + 2''k'' are valid arguments of for all integers , because adding 2''k'' radian or ''k''⋅360° to corresponds to "winding" around the origin counter-clock-wise by Turn (geometry), 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., $-\pi < \varphi \le \pi$ or $0 \le \varphi < 2\pi.$ These regions, where the argument of is uniquely determined are called principal branch, ''branches'' of the argument function. Euler's formula connects the trigonometric functions sine and cosine to the complex exponential: :$e^ = \cos \varphi + i\sin \varphi .$ Using this formula, and again the periodicity, the following identities hold: :$\beginz& = & r \left \left(\cos \varphi + i \sin \varphi\right\right) \\ & = & r \left \left(\cos\left(\varphi + 2k\pi\right) + i \sin\left(\varphi + 2k\pi\right)\right\right) \\ & = & r e^ \\ & = & e^ e^ \\ & = & e^ = e^, \end$ 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 :$a_k = \ln \left(r\right) + i \left( \varphi + 2 k \pi \right),\quad$ for arbitrary integers . Taking such that $\varphi + 2 k \pi$ 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 Exponentiation#Failure of power and logarithm identities, 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. Dropping the range restrictions on the argument makes the relations "argument of ", and consequently the "logarithm of ", multi-valued functions.

## 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, the logarithm of a matrix is the (multi-valued) inverse function of the matrix exponential. Another example is the p-adic logarithm function, ''p''-adic logarithm, the inverse function of the p-adic exponential function, ''p''-adic exponential. Both are defined via Taylor series analogous to the real case. In the context of differential geometry, the exponential map (Riemannian geometry), exponential map maps the tangent space at a point of a differentiable manifold, manifold to a neighborhood (mathematics), neighborhood of that point. Its inverse is also called the logarithmic (or log) map. In the context of finite groups exponentiation is given by repeatedly multiplying one group element with itself. The
discrete logarithm In mathematics, for given real numbers ''a'' and ''b'', the logarithm log''b'' ''a'' is a number ''x'' such that . Analogously, in any group ''G'', powers ''b'k'' can be defined for all integers ''k'', and the discrete logarithm log''b''& ...
is the integer ''n'' 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, such as for example in the Diffie–Hellman key exchange, a routine that allows secure exchanges of cryptography, cryptographic 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. Further logarithm-like inverse functions include the ''double logarithm'' ln(ln(''x'')), the ''super-logarithm, super- or hyper-4-logarithm'' (a slight variation of which is called iterated logarithm in computer science), the Lambert W function, and the logit. They are the inverse functions of the double exponential function, tetration, of , and of the logistic function, respectively.

## Related concepts

From the perspective of group theory, the identity expresses a group isomorphism between positive real number, 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 (Lebesgue measure) ''dx'' 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, called the probability semiring; this is in fact a semifield. The logarithm then takes multiplication to addition (log multiplication), and takes addition to log addition (LogSumExp), giving an isomorphism of semirings between the probability semiring and the log semiring. logarithmic form, Logarithmic one-forms appear in complex analysis and algebraic geometry as differential forms with logarithmic Pole (complex analysis), poles. The polylogarithm is the function defined by :$\operatorname_s\left(z\right) = \sum_^\infty .$ It is related to the
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , where is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if the ...
by . Moreover, equals the Riemann zeta function .