Many mathematical and physical relationships are functionally dependent on high-order variables.
Consider the Stefan-Boltzmann law, which relates the power (j*) emitted by a black body to temperature (T).
<equation contenteditable="false">$j^*= σT^4$
On a standard graph, this equation can be quite unwieldy. The fourth-degree dependence on temperature means that power increases extremely quickly. The fact that the rate is ever-increasing (and steeply so) means that changing scale is of little help in making the graph easier to interpret.
For very steep functions, it is possible to plot points more smoothly while retaining the integrity of the data: one can use a graph with a logarithmic scale (Figure 1).
The primary difference between the logarithmic and linear scales is that, while the difference in value between linear points of equal distance remains constant (that is, if the space from 0 to 1 on the scale is 1cm on the page, the distance from 1 to 2, 2 to 3, etc., will be the same), the difference in value between points on a logarithmic scale will change exponentially. A logarithmic scale will start at a certain power of 10, and with every unit will increase by a power of 10.
Thus, if one wanted to convert a linear scale (with values 0-5) to a logarithmic scale, one option would be to replace 0, 1, 2, 3, 4, and 5 with 0.001, 0.01, 0.1, 1, 10, and 100, respectively. Between each major value on the logarithmic scale, the hashmarks become increasingly closer together with increasing value. For example, in the space between 1 and 10, the 8 and 9 are much closer together than the 2 and 3.
The advantages of using a logarithmic scale are twofold. Firstly, doing so allows one to plot a very large range of data without losing the shape of the graph. Secondly, it allows one to interpolate at any point on the plot, regardless of the range of the graph (Figure 3). Similar data plotted on a linear scale is less clear (Figure 2).
A key point about using logarithmic graphs to solve problems is that they expand scales to the point at which large ranges of data make more sense. In logarithms, the product of numbers is the sum of their logarithms. In the equation mentioned above, plotting j vs. T would generate the expected curve, but the scale would be such that minute changes go unnoticed and the large scale effects of the relationship dominate the graph—it is so big that the "interesting areas" won't fit on the paper on a readable scale.
Taking logarithms, however, one ends with