What is the value of $[\log_{10}(5\log_{10}100)]^2$?
$\textbf{(A)}\ \log_{10}50 \qquad \textbf{(B)}\ 25\qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 2\qquad \textbf{(E)}\ 1$
$[\log_{10}(5\log_{10}100)]^2$
Since $10^2 = 100$, we have $\log_{10} 100 = 2$, so:
$[\log_{10}(5\cdot 2)]^2$
Multiply:
$[\log_{10}(10)]^2$
Since $10^1 = 10$, we have $\log_{10} 10 = 1$, so:
$[1]^2$
$1$
$\fbox{E}$.