Long Log

$\begin{aligned} \log_{\frac {\sqrt x}x} \left(\log_x \sqrt x\right) & = \frac {\frac x2}2 \\ \log_{x^{-\frac 12}} \left(\log_x x^\frac 12\right) & = \frac x4 \\ \log_{x^{-\frac 12}} \left(\frac 12 \log_x x\right) & = \frac x4 \\ \log_{x^{-\frac 12}} \left(\frac 12\right) & = \frac x4 \\ \frac {\log_x \frac 12}{\log_x x^{-\frac 12}} & = \frac x4 \\ 2\log_x 2 & = \frac x4 \\ \implies x & = 8 \log_x 2 \\ x^x & = 2^8 = 4^4 \\ \implies x & = \boxed{4} \end{aligned}$