Answer
$x=4$
Work Step by Step
$\log_2x^{\log_2x}=4$
$(\log_2x)(\log_2x)=4$
$(\log_2x)^2=4$
$2\cdot\log_2x=4$
$\log_2x=\dfrac{4}{2}$
$\log_2x=2$
$x=2^2$
$x=4$
Verifying the answer:
$\log_24^{\log_24}=4$
$2^2=4$
$4=4\checkmark$