Logarithmic scale

A logarithmic scale (or log scale) is a special way to show numbers that can be very different in size, like comparing tiny insects to huge whales. It helps us see and compare these big differences more easily.

Unlike a regular scale where each step is the same size, a logarithmic scale works by multiplying by a set number each time. For example, instead of counting 1, 2, 3, 4, we might count 10, 100, 1000, and so on. This makes big numbers easier to compare.

Because it multiplies instead of adding, a logarithmic scale is not straight like a ruler. Numbers that look the same distance apart on this scale might be very different in real life, like 10, 100, and 1000. This helps show how things grow really fast, like exponential growth, which is often drawn on these special graphs.

Logarithmic scales were very useful in the past with tools like the slide rule, which helped people do hard math without calculators. Today, we still use them in science and engineering to show big changes in things like sound, light, and earthquakes.

Common uses

The markings on slide rules use a special scale called a log scale to help with multiplying or dividing numbers by adding or subtracting lengths.

Logarithmic scales are used in many areas where numbers can vary a lot. Some common examples include:

• Economic growth
• Richter magnitude scale and moment magnitude scale for measuring the strength of earthquakes
• Sound level, measured in units called decibel
• Frequency level in music, using units like octave
• pH for measuring acidity
• Stellar magnitude scale for measuring how bright stars are

Our senses, like hearing, sometimes work in a way that is similar to logarithmic scales, which is why they are useful for showing certain kinds of information.

Graphic representation

The top left graph uses a regular scale for both the X- and Y-axes, showing values from 0 to 10. The bottom left graph uses a special scale called a base-10 log scale for the Y-axis, showing values from 0.1 to 1000.

The top right graph uses a log-10 scale only for the X-axis, and the bottom right graph uses a log-10 scale for both the X and Y-axes.

Using a log scale helps when the data:

• has a very wide range of values, because it makes the data easier to see and compare;
• follows patterns that grow quickly, like exponential laws or power laws, because these appear as straight lines.

A slide rule uses logarithmic scales, and nomograms often do too. The geometric mean of two numbers sits exactly between them on a log scale. Before computers could draw graphs, scientists often used logarithmic graph paper.

Log–log plots

Main article: Log–log plot

If both the vertical and horizontal axes of a graph use a logarithmic scale, the graph is called a log–log plot.

Semi-logarithmic plots

Main article: Semi-log plot

If only one axis—either the vertical (ordinate) or horizontal (abscissa)—uses a logarithmic scale, the graph is called a semi-logarithmic plot.

Extensions

A special change can be made for negative values.

• bel, decibel
• neper

Units of frequency level

• decade, decidecade, savart
• octave, tone, semitone, cent

Table of examples