Answer
{$\frac{3}{2}$}
Work Step by Step
For all positive numbers $a$, where $a\ne1$, and all positive numbers $x$, we know that $y=log_{a}x$ means the same as $x=a^{y}$.
Therefore, $log_{4}\sqrt 64=x$ means the same as $4^{x}=\sqrt 64$.
We know that $x=\frac{3}{2}$, because $4^{\frac{3}{2}}=(4^{3})^{\frac{1}{2}}=64^{\frac{1}{2}}=\sqrt 64$.