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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 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]}

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

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

To find the slope of the plot, two points are selected on the *x*-axis, say *x*_{1} and *x*_{2}. Using the above equation:

and

The slope *m* is found taking the difference:

where *F*_{1} is shorthand for *F* ( *x*_{1} ) and the same for *F*_{2}. 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:

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* (*x*_{0}, *F*_{0}), where *F*_{0} is shorthand for *F*(*x*_{0}), somewhere on the straight line in the above graph, and further some other *arbitrary point* (*x*_{1}, *F*_{1}) on the same graph. Then from the slope formula above:

which leads to

Notice that 10^{log10(F1)} = *F*_{1}. Therefore, the logs can be inverted to find:

or

which means that

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 (*F*_{0}, *x*_{0}) and (*F*_{1}, *x*_{1}) will have the function:

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

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

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.