Question 15
Positive real numbers $a$ and $b$ have the property that
$$
\begin{aligned} &\sqrt{\log a} + \sqrt{\log b} \\ &\quad {}+ \log \sqrt{a} + \log \sqrt{b} = 100 \end{aligned}
$$
and all four terms on the left are positive integers, where $\log$ denotes the base $10$ logarithm. What is $ab?$
$$
\begin{aligned} &\sqrt{\log a} + \sqrt{\log b} \\ &\quad {}+ \log \sqrt{a} + \log \sqrt{b} = 100 \end{aligned}
$$
and all four terms on the left are positive integers, where $\log$ denotes the base $10$ logarithm. What is $ab?$