Main Concept

The sizes of the massive bodies in the Solar System (Sun, planets, planetoids, moons, asteroids, etc.) vary widely over many orders of magnitude. Furthermore, there are far more of the smaller objects than there are of the bigger ones. This is an example of a common natural phenomenon which can be roughly described by the exponential distribution:

A set of events $\left\{E\right\}$ satisfies an exponential distribution when the probability of an event E having a certain measure x is roughly proportional to $e^{-\mathrm{\λ} x}$ for some $\mathrm{\λ}\>0$.

In order to analyze a series of objects satisfying an exponential distribution, it can be better to plot the occurrences of each object on a scale where the objects are likely to be more equally spaced. To do this, we use a logarithmic scale, where an axis represents the value of $\ln \left(x\right)$ as opposed to a linear scale, where an axis represents $x$ itself. Using a logarithmic scale also has the advantage that the ratio between the highest and lowest values of $\ln \left(x\right)$ for objects in the set is much less than the corresponding ratio between the highest and lowest values of $x$ itself, making it easier to compare the different values.

Plotting on a linear vs. logarithmic scale

Below is a plot of the sizes of planets and the Sun in the solar system. On the vertical axis the points are arranged using a linear scale, and you can see that points are all bunched up in one small region near the bottom of the plot, and there is an outlier point (the Sun) near the top. In contrast, the horizontal axis uses a logarithmic scale, and you can see that the points are more evenly spread out.     Because the points are more spread out, there are more regular distances between points of interest. It is therefore more instructive to view this data using a logarithmic scale, or for example in a view window with an exponentially growing zoom.   In the following diagram, as you use the slider to adjust the zoom, you will notice that with linear scaling, the planets are all visible only for quite a small subset of the scale values. If you switch to logarithmic scaling, the planets are visible for a greater proportion of the possible slider values.

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