[Geometry]
Problem: Determine all values of $x$ for which $(\sqrt{x})^{\log _{10} x}=100$.

Solution: Using rules for manipulating logarithms,

$$
\begin{aligned}
(\sqrt{x})^{\log _{10} x} & =100 \\
\log _{10}\left((\sqrt{x})^{\log _{10} x}\right) & =\log _{10} 100 \\
\left(\log _{10} x\right)\left(\log _{10} \sqrt{x}\right) & =2 \\
\left(\log _{10} x\right)\left(\log _{10} x^{\frac{1}{2}}\right) & =2 \\
\left(\log _{10} x\right)\left(\frac{1}{2} \log _{10} x\right) & =2 \\
\left(\log _{10} x\right)^{2} & =4 \\
\log _{10} x & = \pm 2 \\
x & =10^{ \pm 2}
\end{aligned}
$$

Therefore, $x=100$ or $x=\frac{1}{100}$.

(We can check by substitution that each is indeed a solution.)

Final answer: $100,\frac{1}{100}$