AMC 12 2010b Problem 12

For what value of $x$ does

$$\log_{\sqrt{2}}\sqrt{x}+\log_{2}{x}+\log_{4}{x^2}+\log_{8}{x^3}+\log_{16}{x^4}=40?$$

\textbf{(A)}\ 8 \textbf{(B)}\ 16 \textbf{(C)}\ 32 \textbf{(D)}\ 256 \textbf{(E)}\ 1024