A logarithmic scale is a nonlinear scale used for a large range of positive multiples of some quantity. Common uses include earthquake strength, sound loudness, light intensity, and pH of solutions.
It is based on orders of magnitude, rather than a standard linear scale, so the value represented by each equidistant mark on the scale is the value at the previous mark multiplied by a constant.
Logarithmic scales are also used in slide rules 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:
The following are examples of commonly used logarithmic scales, where a larger quantity results in a lower (or negative) value:
Some of our senses operate in a logarithmic fashion (Weber-Fechner law), which makes logarithmic scales for these input quantities especially appropriate. In particular our sense of hearing 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.^{[1]} It can also be used for geographical purposes like for measuring the speed of earthquakes.
The top left graph is linear in the X and Y axis, 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:
A slide rule has logarithmic scales, and nomograms often employ logarithmic scales. The geometric mean of two numbers is midway between the numbers. Before the advent of computer graphics, logarithmic graph paper was a commonly used scientific tool.
If both the vertical and horizontal axes of a plot are scaled logarithmically, the plot is referred to as a log-log plot.
If only the ordinate or abscissa is scaled logarithmically, the plot is referred to as a semi-logarithmic plot.
A logarithmic unit is a unit 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 of logarithmic units include units of data storage capacity (bit, byte), of information and information entropy (nat, shannon, ban), signal level (decibel, bel, neper). Logarithmic frequency quantities are used in electronics (decade, octave) and for music pitch intervals (octave, semitone, cent, etc.). Other logarithmic scale units include the Richter magnitude scale point.
Unit | Base of logarithm | Underlying quantity | Interpretation |
---|---|---|---|
bit | 2 | number of possible messages | quantity of information |
byte | 2^{8} = 256 | number of possible messages | quantity of information |
decibel | 10^{(1/10)} ≈ 1.259 | any power quantity (sound power, for example) | sound power level (for example) |
decibel | 10^{(1/20)} ≈ 1.122 | any root-power quantity (sound pressure, for example) | sound pressure level (for example) |
semitone | 2^{(1/12)} ≈ 1.059 | frequency of sound | pitch interval |
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.^{[]}
The motivation behind the concept of logarithmic units is that defining a quantity on a logarithmic scale in terms of a logarithm to a specific base amounts to making a (totally arbitrary) choice of a unit of measurement for that quantity, one that corresponds to the specific (and equally arbitrary) logarithm base that was selected. Due to the identity
the logarithms of any given number a to two different bases (here b and c) differ only by the constant factor log_{c} b. This constant factor can be considered to represent the conversion factor for converting a numerical representation of the pure (indefinite) logarithmic quantity Log(a) from one arbitrary unit of measurement (the [log c] unit) to another (the [log b] unit), since
For example, Boltzmann's standard definition of entropy S = k ln W (where W is the number of ways of arranging a system and k is Boltzmann's constant) can also be written more simply as just S = Log(W), where "Log" here denotes the indefinite logarithm, and we let k = [log e]; that is, we identify the physical entropy unit k with the mathematical unit [log e]. This identity works because
Thus, we can interpret Boltzmann's constant as being simply the expression (in terms of more standard physical units) of the abstract logarithmic unit [log e] that is needed to convert the dimensionless pure-number quantity ln W (which uses an arbitrary choice of base, namely e) to the more fundamental pure logarithmic quantity Log(W), which implies no particular choice of base, and thus no particular choice of physical unit for measuring entropy.