There are some very nice insights in here. (As usual from Terry Tao.) For instance (here I'm simplifying a bit; if you want a more accurate version, go read TT's article): if you've got two random variables X and Y, and X approximately obeys a power-law distribution, and Y is independent of X and has some other not-terribly-demanding properties, then XY also obeys a power-law distribution, to the same degree of accuracy or (often) better. So, things made out of products of random variables (e.g., because they arise from an exponential growth process applied to some starting random thing) tend to have this sort of distribution, which is where things like Benford's law (first digits of positive "random" numbers tend to occur unevenly, with k occuring log(1+1/k) of the time) come from.
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There are some very nice insights in here. (As usual from Terry Tao.) For instance (here I'm simplifying a bit; if you want a more accurate version, go read TT's article): if you've got two random variables X and Y, and X approximately obeys a power-law distribution, and Y is independent of X and has some other not-terribly-demanding properties, then XY also obeys a power-law distribution, to the same degree of accuracy or (often) better. So, things made out of products of random variables (e.g., because they arise from an exponential growth process applied to some starting random thing) tend to have this sort of distribution, which is where things like Benford's law (first digits of positive "random" numbers tend to occur unevenly, with k occuring log(1+1/k) of the time) come from.