MathNet 2018-part-07 — 177 problems (EN–VI) Question 129

Let $a$ and $b$ be real numbers greater than $1$ such that $a b = 100$. The maximum possible value of $a^{\left(\log_{10} b\right)^2}$ can be written in the form $10^{x}$ for some real number $x$. Find $x$.