Log-log plot

In science and engineering, a log-log graph or log-log plot is a two-dimensional graph of numerical data that uses logarithmic scales on both the horizontal and vertical axes. Because of the nonlinear scaling of the axes, a function of the form $y=ax^b$ will appear as a straight line on a log-log graph, in which b will be the slope of the line (gradient) and a will be the y value corresponding to x = 1.^[1]

1 Equations
- 1.1 Slope of a log-log plot
- 1.2 Finding the function from the log-log plot
2 Applications
3 References

Equations

The equation for a line on a log-log scale would be:

$\log_{10}(F(x)) = m \log_{10}(x) %2B b,$

$F(x) = (x^m)(10^b),$

where m is the slope and b is the intercept point on the log plot.

Slope of a log-log plot

To find the slope of the plot, two points are selected on the x-axis, say x₁ and x₂. Using the above equation:

$\log[F (x_1)] = m \log (x_1) %2B b, \,$

and

$\mathrm {log}[F (x_2)] = m \log (x_2) %2B b. \,$

The slope m is found taking the difference:

$m = \frac { \mathrm {log} (F_2) - \mathrm {log} (F_1)} { \log(x_2) - \log(x_1) } = \frac {\log (F_2/F_1)}{\log(x_2/x_1)}, \,$

where F₁ is shorthand for F ( x₁ ) and the same for F₂. The figure at right illustrates the formula. Notice that the slope in the example of the figure is negative. The formula also provides a negative slope, as can be seen from the following property of the logarithm:

$\log(x_1/x_2) = -\log(x_2/x_1). \,$

Finding the function from the log-log plot

The above procedure now is reversed to find the form of the function F(x) using its (assumed) known log-log plot. To find the function F, pick some fixed point (x₀, F₀), where F₀ is shorthand for F(x₀), somewhere on the straight line in the above graph, and further some other arbitrary point (x₁, F₁) on the same graph. Then from the slope formula above:

$m = \frac {\log (F_1 / F_0)}{\log(x_1 / x_0)}$

which leads to

$\log(F_1 / F_0) = m \log(x_1 / x_0) = \log[(x_1 / x_0)^m ]. \,$

Notice that 10^{log₁₀(F₁)} = F₁. Therefore, the logs can be inverted to find:

$\frac{F_1}{F_0} = \left(\frac{x_1}{x_0}\right)^m$

$F_1 = \frac{F_0}{x_0^m} \,\, x^m, \,$

which means that

$F(x) = \mathrm{constant}\cdot x^m.$

In other words, F is proportional to x to the power of the slope of the straight line of its log-log graph. Specifically, a straight line on a log-log plot containing points (F₀, x₀) and (F₁, x₁) will have the function:

$F(x) = {F_0}\left(\frac{x}{x_0}\right)^\frac {\log (F_1/F_0)}{\log(x_1/x_0)},$

Of course, the inverse is true too: any function of the form

$F(x) = \mathrm{constant} \cdot x^m$

will have a straight line as its log-log graph representation, where the slope of the line is m.

Applications

These graphs are useful when the parameters a and b need to be estimated from numerical data, and can also be used to estimate the fractal dimension of a naturally occurring fractal. Notably, when a probability distribution follows a power law, it will appear as a line on a log-log scale. These graphs are also extremely useful when data are gathered by varying the control variable along an exponential fashion. In this case, log-log graphing will yield a graph that shows the data points as evenly spaced, despite the compression of points at the low end.

References

^ M. Bourne Graphs on Logarithmic and Semi-Logarithmic Paper (www.intmath.com)