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')