What does the Matlab loglog plots do that is useful? If there is a exponential relationship $y = e^x$ and we take the logarithm of this we can see a linear relationship $\ln (y) = x$. So we could plot the logarithm of the y-axis values against the x values. Why does matlab do loglog plots, where the x-axis is also presented in logarithms? I see many plots where this is done. 
Wouldn't using semilogy suffice? Or does using loglog or semilogy actually make no difference for a particular reason?
 A: Besides looking for exponential relationships, one might also want to track power law relationships. As Didier and I pointed out, log-log plots are ideal for the task because they reduce any power law relationship to a linear relationship. The exponent then becomes the slope of the graphic, as seen from
$$y=c x^a \Leftrightarrow \log y = a \log x + \log c$$
Also see this question.
A: I ran into this old question, and I think it needs a little updating.
In communication engineering (and I would think in many other fields too) using $\operatorname{loglog}$ is very common whenever you need to plot a quantity that has a huge dynamic range in both $x$ and $y$ axes. For example, in signal processing one often computes the power spectrum of a signal. The result most likely spans many decades in frequency and many decades in amplitude. $\operatorname{loglog}$ allows you to do be able to see the result, which otherwise would be hard to interpret using any other plotting function.
I think that on a day to day basis this happens way more frequently than attempting to detect power laws. Commercial instruments that perform RF measurements are commonly setup to show both $x$ and $y$ axes in a $\operatorname{loglog}$-like fashion, for the reason I just mentioned.
See this simple example (originally from here), and try to replace loglog with plot, semilogx and semilogy. You'll see that only loglog allows you to see a meaningful plot.
Fs = 1000;
t = 0:(1/Fs):1000;
x = sin(pi*t);
[Pxx, f] = pwelch(x, [], [], [], Fs);
loglog(f, Pxx)
grid on
xlabel('Frequency (Hz)')
ylabel('Magnitude (units^2/Hz)')
title('PSD of Sine Wave')

