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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 scale is
nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
: the numbers 10 and 20, and 60 and 70, are not the same distance apart on a log scale. Rather, the numbers 10 and 100, and 60 and 600 are equally spaced. Thus moving a unit of distance along the scale means the number has been ''multiplied'' by 10 (or some other fixed factor). Often
exponential growth Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a ...
curves are displayed on a log scale, otherwise they would increase too quickly to fit within a small
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
. Another way to think about it is that the ''number of digits'' of the data grows at a constant rate. For example, the numbers 10, 100, 1000, and 10000 are equally spaced on a log scale, because their numbers of digits is going up by 1 each time: 2, 3, 4, and 5 digits. In this way, adding two digits ''multiplies'' the quantity measured on the log scale by a factor of 100.

# Common uses

The markings on slide rules are arranged in a log scale for multiplying or dividing numbers by adding or subtracting lengths on the scales. The following are examples of commonly used logarithmic scales, where a larger quantity results in a higher value: *
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 ...
and moment magnitude scale (MMS) for strength of
earthquakes An earthquake (also known as a quake, tremor or temblor) is the shaking of the surface of the Earth resulting from a sudden release of energy in the Earth's lithosphere that creates seismic waves. Earthquakes can range in intensity, fro ...
and movement in the
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surface ...
* Sound level, with units decibel *
Neper The neper (symbol: Np) is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain and loss of electronic signals. The unit's name is derived from the name of John Napier, the inventor of logarithms. A ...
for amplitude, field and power quantities * Frequency level, with units
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 * ...
, minor second, major second, and
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 refer ...
for the relative pitch of notes in music *
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 transformations. Mathematically, the logit is the i ...
for
odds Odds provide a measure of the likelihood of a particular outcome. They are calculated as the ratio of the number of events that produce that outcome to the number that do not. Odds are commonly used in gambling and statistics. Odds also have ...
in statistics * Palermo Technical Impact Hazard Scale * Logarithmic timeline * Counting
f-stop In optics, the f-number of an optical system such as a camera lens is the ratio of the system's focal length to the diameter of the entrance pupil ("clear aperture").Smith, Warren ''Modern Optical Engineering'', 4th Ed., 2007 McGraw-Hill P ...
s for ratios of photographic exposure * The rule of nines used for rating low
probabilities Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...
* Entropy in
thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws ...
* Information in information theory * Particle size distribution curves of soil The following are examples of commonly used logarithmic scales, where a larger quantity results in a lower (or negative) value: * pH for acidity * Stellar magnitude scale for brightness of stars * Krumbein scale for
particle size Particle size is a notion introduced for comparing dimensions of solid particles ('' flecks''), liquid particles (''droplets''), or gaseous particles ('' bubbles''). The notion of particle size applies to particles in colloids, in ecology, in gr ...
in
geology Geology () is a branch of natural science concerned with Earth and other astronomical objects, the features or rocks of which it is composed, and the processes by which they change over time. Modern geology significantly overlaps all other Ear ...
*
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 by transparent samples Some of our
sense A sense is a biological system used by an organism for sensation, the process of gathering information about the world through the detection of stimuli. (For example, in the human body, the brain which is part of the central nervous system r ...
s operate in a logarithmic fashion (
Weber–Fechner law The Weber–Fechner laws are two related hypotheses in the field of psychophysics, known as Weber's law and Fechner's law. Both laws relate to human perception, more specifically the relation between the actual change in a physical stimulus and ...
), which makes logarithmic scales for these input quantities especially appropriate. In particular, our sense of
hearing Hearing, or auditory perception, is the ability to perceive sounds through an organ, such as an ear, by detecting vibrations as periodic changes in the pressure of a surrounding medium. The academic field concerned with hearing is auditory ...
perceives equal ratios of frequencies as equal differences in pitch. In addition, studies of young children in an isolated tribe have shown logarithmic scales to be the most natural display of numbers in some cultures.

# Graphic representation

The top left graph is linear in the X and Y axes, and the Y-axis ranges from 0 to 10. A base-10 log scale is used for the Y axis of the bottom left graph, and the Y axis ranges from 0.1 to 1,000. The top right graph uses a log-10 scale for just the X axis, and the bottom right graph uses a log-10 scale for both the X axis and the Y axis. Presentation of data on a logarithmic scale can be helpful when the data: * covers a large range of values, since the use of the logarithms of the values rather than the actual values reduces a wide range to a more manageable size; * may contain exponential laws or
power law In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one q ...
s, since these will show up as straight lines. A slide rule has logarithmic scales, and nomograms often employ logarithmic scales. The
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 ...
of two numbers is midway between the numbers. Before the advent of computer graphics, logarithmic
graph paper Graph paper, coordinate paper, grid paper, or squared paper is writing paper that is printed with fine lines making up a regular grid. The lines are often used as guides for plotting graphs of functions or experimental data and drawing curves. ...
was a commonly used scientific tool.

## Log–log plots

If both the vertical and horizontal axes of a plot are scaled logarithmically, the plot is referred to as a log–log plot.

## Semi-logarithmic plots

If only the ordinate or abscissa is scaled logarithmically, the plot is referred to as a semi-logarithmic plot.

## Extensions

A modified log transform can be defined for negative input (''y''<0) and to avoid the singularity for zero input (''y''=0) so as to produce symmetric log plots: :$Y=\sgn\left(y\right)\cdot\log_\left(1+, y/C, \right)$ for a constant ''C''=1/ln(10).

# Logarithmic units

A logarithmic unit 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'' (al ...
that can be used to express a quantity ( physical or mathematical) on a logarithmic scale, that is, as being proportional to the value of a logarithm function applied to the ratio of the quantity and a reference quantity of the same type. The choice of unit generally indicates the type of quantity and the base of the logarithm.

## Examples

Examples of logarithmic units include units of data storage capacity ( bit, byte), of information and information entropy ( nat, shannon, ban), and of signal level ( decibel, bel,
neper The neper (symbol: Np) is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain and loss of electronic signals. The unit's name is derived from the name of John Napier, the inventor of logarithms. A ...
). Logarithmic frequency quantities are used in electronics (
decade A decade () is a period of ten years. Decades may describe any ten-year period, such as those of a person's life, or refer to specific groupings of calendar years. Usage Any period of ten years is a "decade". For example, the statement that "du ...
,
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 refer ...
) and for music pitch intervals (
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 refer ...
, semitone,
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 * ...
, etc.). Other logarithmic scale units include 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 ...
point. In addition, several industrial measures are logarithmic, such as standard values for
resistors A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active ...
, the American wire gauge, the
Birmingham gauge The Birmingham gauge is a wire gauge system, and is also used to specify thickness or diameter of hypodermic needles and tube products. Terminology Birmingham gauge is also known as the Stubs Iron Wire Gauge or Birmingham Wire Gauge. It is not ...
used for wire and needles, and so on.

## Units of information

* bit, byte * hartley * nat * shannon

## Units of level or level difference

* bel, decibel *
neper The neper (symbol: Np) is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain and loss of electronic signals. The unit's name is derived from the name of John Napier, the inventor of logarithms. A ...

## Units of frequency interval

*
decade A decade () is a period of ten years. Decades may describe any ten-year period, such as those of a person's life, or refer to specific groupings of calendar years. Usage Any period of ten years is a "decade". For example, the statement that "du ...
,
decidecade A one-third octave is a logarithmic unit of frequency ratio equal to either one third of an octave (1200/3 = 400 cents: major third) or one tenth of a decade (3986.31/10 = 398.631 cents: M3 ). An alternative (unambiguous) term for one tenth o ...
, savart *
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 refer ...
, tone, semitone,
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 * ...

## Table of examples

The two definitions of a decibel are equivalent, because a ratio of power quantities is equal to the square of the corresponding ratio of root-power quantities.

*
Alexander Graham Bell Alexander Graham Bell (, born Alexander Bell; March 3, 1847 – August 2, 1922) was a Scottish-born inventor, scientist and engineer who is credited with patenting the first practical telephone. He also co-founded the American Telephone and Te ...
* Bode plot *
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 ...
(arithmetic mean in logscale) *
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 Ioan ...
*
Level (logarithmic quantity) In science and engineering, a power level and a field level (also called a root-power level) are logarithmic measures of certain quantities referenced to a standard reference value of the same type. * A ''power level'' is a logarithmic quantity ...
* Logarithm * Logarithmic mean * Log semiring * Preferred number * Semi-log plot

## Scale

*
Order of magnitude An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic dis ...

## Applications

* Entropy *
Entropy (information theory) In information theory, the entropy of a random variable is the average level of "information", "surprise", or "uncertainty" inherent to the variable's possible outcomes. Given a discrete random variable X, which takes values in the alphabet ...
* pH *
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 ...