Logarithmic scale

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LinLinScale.png
LinLogScale.png
LogLinScale.png
LogLogScale.svg
Various scales: lin-lin, lin-log, log-lin and log-log. Plotted graphs are: y=x (green), y=10x(red), y=log(x) (blue).

A logarithmic scale is a scale of measurement that uses the logarithm of a physical quantity instead of the quantity itself. On a logarithmic scale, each tick mark on the scale is the previous tick mark multiplied by some number.

The logarithmic scale can be helpful when the data covers a large range of values – the logarithm reduces this to a more manageable range.

Some of our senses operate in a logarithmic fashion (multiplying the actual input strength adds a constant to the perceived signal strength, see: Stevens' power law). That makes logarithmic scales for these input quantities especially appropriate. In particular, our sense of hearing perceives equal multiples of frequencies as equal differences in pitch.

On most logarithmic scales, small multiples (or ratios) of the underlying quantity correspond to small (possibly negative) values of the logarithmic measure.

Examples[change | edit source]

Well-known examples of such scales are:

Some logarithmic scales were designed such that large values (or ratios) of the underlying quantity correspond to small values of the logarithmic measure. Examples of such scales are:

Graphic representation[change | edit source]

A logarithmic scale is also a graphical scale on one or both sides of a graph where a number x is printed at a distance c·log(x) from the point marked with the number 1. A slide rule has logarithmic scales, and nomograms often employ logarithmic scales. On a logarithmic scale an equal difference in order of magnitude is represented by an equal distance. The geometric mean of two numbers is midway between the numbers.

Logarithmic graph paper, before the advent of computer graphics, was a basic scientific tool. Plots on paper with one log scale can show up exponential laws, and on log-log paper power laws, as straight lines (see semilog graph, log-log graph).