The Mathematics behind Benford's Law

The mathematical formula representation of Benford's Distribution is:

P(d) = log(d+1) - log(d) = log(1+(1/d)).

This can be further explained as:

The Probability P of digit(d=1) existing in the first place is log(1+(1/1)) = log(2)

P(1) = .301

and the Probability of the first digit being a 9:

P(9) = log((1+(1/9)) = log(1.111) = .046