Let $r$ be the result of doubling both the base and exponent of $a^b$, and $b$ does not equal to $0$. 
If $r$ equals the product of $a^b$ by $x^b$, then $x$ equals:
$\textbf{(A)}\ a \qquad \textbf{(B)}\ 2a \qquad \textbf{(C)}\ 4a \qquad \textbf{(D)}\ 2\qquad \textbf{(E)}\ 4$
From the problem, $r = (2a)^{2b}$, so
\[(2a)^{2b} = a^b \cdot x^b\]
\[(4a^2)^b = (ax)^b\]
\[4a^2 = ax\]
\[x = 4a\]
Thus, the answer is $\boxed{\textbf{(C)}}$.