What does the slope of a log-log plot represent? I have been asked to plot the relationship of $log(I)$ versus $log(R)$, where $I$ is the moment of inertia of an object and $R$ is the radius used to calculate $I$. What does the slope of this plot represent?
Also, what would the vertical axis intercept of this plot represent?

$$
y = ax^k
$$
taking logs we have
$$
\log y = k \log x + \log a 
$$
if we relabel as
$$
\bar{y} = k \bar{x} + c
$$
we should see that the gradient of the last equation i.e. the $k$, maps to be the gradient in the log-log plot which in turn maps to being the exponent of the original equation.
So in short the gradient of the log-log determines if the original equation is a power law one, and if the gradient indeed does not change then we can assume that the exponent of the power law equation is the gradient.
The Intercept $c = \log a$ which is basically the coefficient of the power law.
