Numer of binary digits of a power of ten (from Weissman's “An Illustrated Theory of Numbers”)

I have been working through the following exercise (chapter $0$, number $17$, in Weissman's An Illustrated Theory of Numbers):

How many bits does the binary expansion of $10^{500}$ have?

I know the number of binary digits is $\lfloor 500 \cdot \log_2 10 \rfloor + 1$, so I can get the answer with a calculator.

Could I know how many bits the binary expansion of $10^{500}$ has, without a calculator or a logarithmic table? Does it help that this exercise considers the number of binary digits of a power of $10$?

It's easy to observe that $10^{500} = 5^{500} \cdot 2^{500}$, so its binary expansion ends with $500$ zeros. Then, I used the following factorization of $5^{500} - 1$:\begin{equation*}\begin{split}5^{500} - 1 &= (5^{250} + 1) (5^{250} - 1) = (5^{250} + 1) (5^{125} + 1) (5^{125} - 1) \\&= (5^{250} + 1) (5^{125} + 1) (5 - 1) (5^{124} + \ldots + 5 + 1).\end{split}\end{equation*}Therefore, the largest power of $2$ dividing $5^{500} - 1$ is $2^4$ and the binary expansion of $5^{500}$ ends with $10001$. But I can't get any further.