If $\log (xy^3) = 1$ and $\log (x^2y) = 1$, what is $\log (xy)$?
$\mathrm{(A)}\ -\frac 12  \qquad\mathrm{(B)}\ 0  \qquad\mathrm{(C)}\ \frac 12 \qquad\mathrm{(D)}\ \frac 35  \qquad\mathrm{(E)}\ 1$
Since 
\begin{align*} &\log(xy) +2\log y = 1  \\ \log(xy) + \log x = 1 \quad \Longrightarrow \quad &2\log(xy) + 2\log x = 2 \end{align*}
Summing gives 
\[3\log(xy) + 2\log y + 2\log x = 3 \Longrightarrow 5\log(xy) = 3\]
Hence $\log (xy) = \frac 35 \Rightarrow \mathrm{(D)}$.
It is not difficult to find $x = 10^{\frac{2}{5}}, y = 10^{\frac{1}{5}}$.