[rflrob]

My favorite explanation of it is that if there is a distribution to the numbers, then that distribution should hold no matter what base you're working in (for natural things, after all, there's nothing special about base 10), and Benfords law can be shown to be a) a law that satisfies this base-independent property, and b) the only law that does so.

[jules]

s/base/units/

[jules]

Seeing that people are downvoting this, perhaps it wasn't obvious enough.

Proof that the starting digit in numbers is not base invariant:

In base 10 not all numbers start with 1. In base 2 all numbers start with 1. Hence, the distribution is not base invariant. QED.

For an explanation why you get the right thing if you substitute "units", I refer you to the Wikipedia page on Benford's Law. http://en.wikipedia.org/wiki/Benfords_law