What's going on? Well, suppose a distribution actually exists for numbers like the lengths of rivers. Clearly it can't be related to the units used to measure the length, so multiplying by a constant can't change matters. So it has to be logarithmic.

If you know any probability, you realise that the above paragraph is meaningless! There is no unbounded uniform distribution, which would seem to work best with the argument. And you cannot prove anything about the distribution if it doesn't exist! Nonetheless, people have managed to prove some versions of Benford's Law, by making "reasonable" assumptions. And empirically, it works!