# What are examples of legitimate usage of logarithmic scale when drawing a chart?

Quite often a chart is drawn using logarithmic scale for one axis (usually the the y-axis). This is often used for abuse when presenting information - logarithmic scale alters greatly how the values are plotted on the plane.

Still I guess there are legitimate cases when using logarithmic scale for one axis which represents a value which is linear in nature (not earthquake magnitude which is already logarithmic itself).

What are examples of such legitimate cases where using logarithmic scale allows for better analysis of plotted data?

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What do you consider an illegitimate case or abuse of the logarithmic scale? Certainly if one is visualizing a power law or exponential growth/decay then a [semi-]logarithmic scale makes it easier to see. – Rahul Apr 12 '12 at 6:08
@Rahul Narain: Well, plotting an exponential decay process on log-scale and then presenting it as if it was linear scale is one example of abuse. – sharptooth Apr 12 '12 at 6:19
I feel that the abuse there lies more in misrepresenting the scale of the graph rather than in the use of a log scale itself. – Rahul Apr 12 '12 at 6:35

## 3 Answers

On a log scale a relationship of the form $x \mapsto x^{\alpha}$ shows up as a straight line (of slope $\alpha$) when you plot $\log x^{\alpha}$ against $\log x$. This is useful with many data sets such as frequency responses in engineering, as you can easily estimate constants just by looking at their graph.

See Darrell Huff's 1954 classic "How to Lie with Statistics" for nice examples of misleading scales.

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There are many answers. Here is one: Many human sensory modes are naturally logarithmic. For example, consider three lamps $A$, $B$, and $C$ of intensities $a$, $a+d$ , and $a+2d$. $C$ will not seem as much brighter compared with $B$ as $B$ does compared with $A$. To get $B$ to appear to be midway in brighness between $A$ and $B$ you need the intensities to be in geometric sequence, say $a$, $ad$, and $ad^2$. You can see this if you have a 50–100–150-watt three-way bulb; the bulb seems to get a lot brighter going from 50 to 100 watts than it does going from 100 to 150 watts. To get the differences to seem the same, the brightest setting would need to be closer to 200 watts, or the middle setting closer to 85 watts.

Now suppose you are making a chart of lamp brightness, and you have three lamps with intensities of 10, 100, and 190 candelas. An ordinary linear chart will give the impression that lamp $C$ is as much brighter, compared with $B$, than $B$ is compared with $A$; both differences are 90 candelas. But such a chart would be very misleading. Actually $B$ and $C$ will appear rather similar, and $A$ will appear much dimmer than both: $B$ is ten times as bright as $A$, but $C$ is less than twice as bright as $A$. A logarithmic axis will present the perceived intensities correctly.

The decibel scale of sound intensity is logarithmic for similar reasons. Auditory pitch is similarly logarithmic: the perceived difference between B-flats on the piano is the same for each consecutive pair, even though the actual frequency difference is twice as great in each case.

The Scoville scale for chili pepper intensity is expressed linearly; it records how much you must dilute the pepper for its peppery taste to drop below the threshhold of sensation. So a mild pepper like an anaheim scores around 1,000, a medium pepper like a serrano around 10,000, and a fiery pepper like a habanero around 100,000. A linear scale suggests that the habanero leaves the serrano in the dust, that compared with a habanero, an anaheim and a serrano are almost indistinguishably bland. But the serrano actually tastes much more like a midpoint between the very hot habanero and the mild anaheim. The scale is misleading, and would have been better expressed logarithmically.

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Variables that give parallel lines on a semilogarithmic scalle are changing at the same rate. For example, plot incidence (cases) and mortality (deaths) for a given diseae to see whether mortality is falling more rapidly than incidence (improved treatment) or whether the lines are declining in parallel (more effective prevention). Plot tuberculosis incidence by race on a semilog scale to see whether rates in Blacks and Whites are declining in parallel (broadly acting control measures) or instead converging (special emphasis to reduce disparities).

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