2004 AMC 12B 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 \qquad\mathrm{(B)}\ 195 \qquad\mathrm{(C)}\ 196 \qquad\mathrm{(D)}\ 197 \qquad\mathrm{(E)}\ 198

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.