What is the value of [ lo g_10 ( 5lo g_10100 )\, ]^2 = ?

Answer: Option A

$$\begin{array}{rl}& {\left[\text{lo}{\text{g}}_{10}\left(\text{5lo}{\text{g}}_{10}100\right)\right]}^2\\ & ={\left[\text{lo}{\text{g}}_{10}\left\{\text{5lo}{\text{g}}_{10}{\left(10\right)}^2\right\}\right]}^2\\ & ={\left[\text{lo}{\text{g}}_{10}\left(5\times 2\right)\right]}^2\\ & ={\left(\text{lo}{\text{g}}_{10}10\right)}^2\\ & =1\end{array}$$