Logarithmic scale

Value

A logarithmic scale is a scale of measurement that gives the logarithm of a physical quantity instead of the quantity itself. This is often done if the underlying quantity can take on a huge range of values; the logarithm reduces this to a more manageable range. Some of our senses operate in a logarithmic fashion (doubling the input strength adds a constant to the subjective signal strength), 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.

Logarithmic scales are either defined for ratios of the underlying quantity, or one has to agree to measure the quantity in fixed units. Deviating from these units means that the logarithmic measure will change by an additive constant. The base of the logarithm also has to be specified.

On most logarithmic scales, small values (or ratios) of the underlying quantity correspond to small (possibly negative) values of the logarithmic measure. Well-known examples of such scales are:

• Richter magnitude scale for strength of earthquakes;
• bel and decibel and neper for acoustic power (loudness) and electric power;
• cent, minor second, major second, and octave for the relative pitch of notes in music;
• logit for odds in statistics;
• Palermo Technical Impact Hazard Scale;
• Logarithmic timeline;
• counting f-stops for ratios of photographic exposure;
• rating low probabilities by the number of 'nines' in the decimal expansion of the probability of their not happening: for example, a system which will fail with a probability of 10^-5 is 99.999% reliable: "five nines".

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:

• pH for acidity;
• stellar magnitude scale for brightness of stars;
• Krumbein scale for grain size in geology.

Graphic representation

A logarithmic scale is also a graphic 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.