2019 AMC 12A 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} \qquad \textbf{(B) } 10^{100} \qquad \textbf{(C) } 10^{144} \qquad \textbf{(D) } 10^{164} \qquad \textbf{(E) } 10^{200}

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