AMC 12 2019a Problem 15

Positive real numbers $a$ and $b$ have the property that $$\sqrt{\log{a}} + \sqrt{\log{b}} + \log \sqrt{a} + \log \sqrt{b} = 100$$

and all four terms on the left are positive integers, where $\log$ denotes the base-$10$ logarithm. What is $ab$?

$\textbf{(A) } 10^{52} \textbf{(B) } 10^{100} \textbf{(C) } 10^{144} \textbf{(D) } 10^{164} \textbf{(E) } 10^{200}$