jmc

Note that $5^n$ has the same number of digits as $5^{n-1}$ if and only if $5^{n-1}$ has a leading digit $1$. Therefore, there are $2004-1401=603$ numbers with leading digit $1$ among the set $\{5^1,5^2,5^3,\cdots 5^{2003}\}.$ However, $5^0$ also starts with $1$, so the answer is $603+1=\boxed{604}$.