AMC 12 2004b Problem 25

Given that $2^{2004}$ is a $604$-digit number whose first digit is $1$, how many elements of the set $S = \{2^0,2^1,2^2,\ldots ,2^{2003}\}$ have a first digit of $4$?

$\mathrm{(A)}\ 194 \mathrm{(B)}\ 195 \mathrm{(C)}\ 196 \mathrm{(D)}\ 197 \mathrm{(E)}\ 198$