Analytic density of the set of primes starting with 1

I think instead of posting my own explanation (which will only lose something in the translation) I'll instead refer you to two very interesting papers (thanks for posting this question, I haven't thought about this stuff in a couple years, and these papers were interesting reads to solve your problem.)

The first (among other things) proves that the density of primes with leading coefficient $k$ is $log_{10}left(frac{k +1}{k}right).$

Prime numbers and the first digit phenomenon
by
Daniel I. A. Cohen* and Talbot M. Katz
in Journal of number theory 18, 261-268 (1984)

The second is a more general statement about first digits. It is

The first digit problem
by
Ralph Raimi
in American Math Monthly vol 83 No 7

Hope this all helps.