7.212. $\frac{\log _{4 \sqrt{x}} 2}{\log _{2 x} 2}+\log _{2 x} 2 \cdot \log _{1 / 2} 2 x=0$.
Solution.

Domain of definition: $\left\{\begin{array}{l}x>0, \\ x \neq \frac{1}{16}, \\ x \neq \frac{1}{2}, \\ x \neq 1 .\end{array}\right.$

Transition to base 2. We have

$\frac{\frac{\log _{2} 2}{\log _{2} 4 \sqrt{x}}}{\frac{\log _{2} 2}{\log _{2} 2 x}}+\frac{\log _{2} 2}{\log _{2} 2 x} \cdot \frac{\log _{2} 2 x}{\log _{2} \frac{1}{2}}=0 \Leftrightarrow \frac{1+\log _{2} x}{2+\frac{1}{2} \log _{2} x}-1=0$,

from which $\log _{2} x=2$ and $x=4$.

Answer: 4.