AMC 12 2022b Problem 16

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 $\log_2{y}$?

$\textbf{(A) }3 \textbf{(B) }4 \textbf{(C) }3+\sqrt{2} \textbf{(D) }4+\sqrt{3} \textbf{(E) }7$