Consider the graphs of $y=2\log{x}$ and $y=\log{2x}$. We may say that:
$\textbf{(A)}\ \text{They do not intersect}\qquad \\ \textbf{(B)}\ \text{They intersect at 1 point only}\qquad \\ \textbf{(C)}\ \text{They intersect at 2 points only} \qquad \\ \textbf{(D)}\ \text{They intersect at a finite number of points but greater than 2} \qquad \\ \textbf{(E)}\ \text{They coincide}$
Substitute $y$ into the other equation.
\[2\log{x} = \log{2x}\]
Using a logarithm property,
\[2\log{x} = \log{2} + \log{x}\]
Subtract both sides by $\log{x}$ to get
\[\log{x}=\log{2}\]
Thus, $x=2$.  Since both $2\log{x}$ and $\log{2x}$ are functions, there is only one corresponding y-value, so the answer is $\boxed{\textbf{(B)}}$.