[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