Logarithmic scale

A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way. As opposed to a linear number line in which every unit of distance corresponds to adding by the same amount, on a logarithmic scale, every unit of length corresponds to multiplying the previous value by the same amount. Hence, such a scale is nonlinear. In nonlinear scale, the numbers 1, 2, 3, 4, 5, and so on would not be equally spaced. Rather, the numbers 10, 100, 1000, 10000, and 100000 would be equally spaced. Likewise, the numbers 2, 4, 8, 16, 32, and so on, would be equally spaced. Often exponential growth curves are displayed on a log scale, otherwise they would increase too quickly to fit within a small graph.

A logarithmic scale from 0.1 to 100
The two logarithmic scales of a slide rule
Semi-log plot of the Internet host count over time shown on a logarithmic scale

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:

A logarithmic scale makes it easy to compare values that cover a large range, such as in this map.
Map of the solar system and distance to Alpha Centauri using a logarithmic scale

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]

Graphic representation
Various scales: lin–lin, lin–log, log–lin, and log–log. Plotted graphs are: y = 10 x (red), y = x (green), y = loge(x) (blue).

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 laws, since these will show up as straight lines.

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.

Log–log plots
Main article: Log–log plot
A log-log plot condensing information that spans more than one order of magnitude along both axes

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
Main article: Semi-log plot

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:[2][3]

for a constant C=1/ln(10).

Logarithmic units

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

Examples of logarithmic units include units of information and information entropy (nat, shannon, ban) and of signal level (decibel, bel, neper). Frequency levels or logarithmic frequency quantities have various units 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.

In addition, several industrial measures are logarithmic, such as standard values for resistors, the American wire gauge, the Birmingham gauge used for wire and needles, and so on.

Units of information

Units of level or level difference
Further information: Level (logarithmic quantity)

Units of frequency level

Table of examples
Unit Base of logarithm Underlying quantity Interpretation
bit 2 number of possible messages quantity of information
byte 28 = 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.

See also

Scale

Applications

References
  1. "Slide Rule Sense: Amazonian Indigenous Culture Demonstrates Universal Mapping Of Number Onto Space". ScienceDaily. 2008-05-30. Retrieved 2008-05-31.
  2. Webber, J Beau W (2012-12-21). "A bi-symmetric log transformation for wide-range data" (PDF). Measurement Science and Technology. IOP Publishing. 24 (2): 027001. doi:10.1088/0957-0233/24/2/027001. ISSN 0957-0233. S2CID 12007380.
  3. "Symlog Demo". Matplotlib 3.4.2 documentation. 2021-05-08. Retrieved 2021-06-22.

Further reading
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