jmc

For a base $b$ representation of $100_{10}$ to have exactly $5$ digits, the largest power of $b$ that is less than $100$ must be $4$. Therefore, we have the requirement that $b^4\le 100<b^5$. We then realize that $b=3$ satisfies this requirement since $3^4<100<3^5$. We also realize that this is the only possible value of $b$ since if $b$ were equal to $2$, $b^5=2^5$ would be less than $100$ and if $b$ were equal to $4$, $b^4=4^4$ would be greater than $100$. Thus, our only solution is $b=\boxed{3}$.