If $x$ and $\log_{10} x$ are real numbers and $\log_{10} x<0$, then:
$\textbf{(A)}\ x<0 \qquad \textbf{(B)}\ -1<x<1 \qquad \textbf{(C)}\ 0<x\le 1 \\ \textbf{(D)}\ -1<x<0 \qquad \textbf{(E)}\ 0<x<1$
Let $\log_{10} x = a$, so $10^a = x$.  Note that $10^0 = 1$ and that $10^a$ increases as $a$ increases.  Since $a  0$, so $0 < x < 1$.  The answer is $\boxed{\textbf{(E)}}$.