The set of $x$-values satisfying the equation $x^{\log_{10} x} = \frac{x^3}{100}$ consists of: 
$\textbf{(A)}\ \frac{1}{10}\qquad\textbf{(B)}\ \text{10, only}\qquad\textbf{(C)}\ \text{100, only}\qquad\textbf{(D)}\ \text{10 or 100, only}\qquad\textbf{(E)}\ \text{more than two real numbers.}$
Taking the base-$x$ logarithm of both sides gives $\log_{10}x=\log_x\frac{x^3}{100}$.
This simplifies to 
\[\log_{10}x=\log_x{x^3} - \log_x{100}\]
\[\log_{10}x+\log_x{100}=3\]
\[\log_{10}x+2 \log_x{10}=3\]
At this point, we substitute $u=\log_{10}x$.
\[u+\frac2u=3\]
\[u^2+2=3u\]
\[u^2-3u+2=0\]
\[(u-2)(u-1)=0\]
\[u\in\{1, 2\}\]
\[x\in\{10, 100\}\]
The answer is $\boxed{\textbf{(D)}}$.