Most people don't understand log-log graphs. Also, taking the log-log is so powerful that even distributions that don't strictly follow a power law can look as if they're approximately following a straight line.
It is much easier to check if the points fall in a straight line than a perfectly draw exponential.
Are you sure that graph is a power law? Are you sure it isn't a thick-tailed hyperbole or some hyperexponential?
being roughly straight on a log-log plot is a necessary
but not sufficient condition for power-law behavior
pp. 24-29 have nice log-log plots of various data sets, along with whether they can plausibly described as power law distributions. 'Wealth', interestingly enough, looks like a pretty straight line, but fails the statistical test badly.
Btw, if you just take logs and run a regression to estimate the exponent, you get a highly biased estimate.
In general, it's much better to do a direct statistical test, although the method described in the paper is somewhat involved.
What I'll say is that what something is is a fuzzy term, although I generally agree what you mean especially about a power law says about the behavior in the tails especially. That comes from thinking a small deviation on the tail is "small" due to optics and not realizing it is deviation on orders of magnitude. That might be what the replier to my comment was referring to. That's why "two graphs" is usually a good answer.
PS Also, I would never fit a straight line on a loglog or semilog graph, nor should anyone ever. People who do that don't understand what least squares is at all.
A polynomial is a power law...with more terms.