Suppose $x$ and $y$ are positive real numbers such that
\[x^y=2^{64}\text{ and }(\log_2{x})^{\log_2{y}}=2^{7}.\]
What is the greatest possible value of $\log_2{y}$?
$\textbf{(A) }3 \qquad \textbf{(B) }4 \qquad \textbf{(C) }3+\sqrt{2} \qquad \textbf{(D) }4+\sqrt{3} \qquad \textbf{(E) }7$
Take the base-two logarithm of both equations to get
\[y\log_2 x = 64\quad\text{and}\quad (\log_2 y)(\log_2\log_2 x) = 7.\]
Now taking the base-two logarithm of the first equation again yields
\[\log_2 y + \log_2\log_2 x = 6.\]
It follows that the real numbers $r:=\log_2 y$ and $s:=\log_2\log_2 x$ satisfy $r+s=6$ and $rs = 7$. Solving this system yields
\[\{\log_2 y,\log_2\log_2 x\}\in\{3-\sqrt 2, 3 + \sqrt 2\}.\]
Thus the largest possible value of $\log_2 y$ is $3+\sqrt 2 \implies \boxed{\textbf {(C)}}$.
cr. djmathman