Logarithmic scale
| 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.
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 (doubling the input strength adds a constant to the subjective signal strength, see: Stevens' power law). That 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.
On most logarithmic scales, small values (or ratios) of the underlying quantity correspond to small (possibly negative) values of the logarithmic measure.
Examples
Well-known examples of such scales are:
- Richter magnitude scale and Moment magnitude scale (MMS) for strength of earthquakes and movement in the earth.
- 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".
- Entropy in thermodynamics.
- Information in information theory.
- Particle Size Distribution curves of soil
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 particle size in geology.
- Kardashev scale for technological advance in physics.
- Absorbance of light by transparent samples.
Graphic representation
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).