But if something can happen at any given scale, the distribution must operate very differently. The distribution can't pick out any favorite size to work at --- some fraction of events must happen at every possible scale. So, if we're looking at the distribution of stellar luminosities, then there will be some stars which are as bright as the Sun, some which are 1% as bright as the Sun, and some which are 100 times as bright as the Sun. And since the star formation process doesn't have any preferred scale, then the number of Solar luminosity stars to stars which are 1% as luminous as the Sun must be about the same as the number of stars which are 100 times as bright as the Sun to the number of Solar luminosity stars. it's perfectly fine for the distribution to prefer dimmer or brighter stars because this doesn't end up introducing a preferred scale --- it just tilts the slope of the distribution (which is a line on a log-log graph).

I'm not a mathematician, so I'm not sure if these statements are true in any rigorous sense, but in practice scale-free processes tend to produce power laws in the universe.

Well worth the 20 minutes.

I recommend reading them through, but slide 39 is the tl;dr.