1. The solution set of the inequality $\sqrt{\log _{2} x-1}+\frac{1}{2} \log _{\frac{1}{2}} x^{3}+2>0$ is
1. $[2,4)$ Detailed explanation: The original inequality is equivalent to $\left\{\begin{array}{l}\sqrt{\log _{2} x-1}-\frac{3}{2} \log _{2} x+\frac{3}{2}+\frac{1}{2}>0 \\ \log _{2} x-1 \geqslant 0\end{array}\right.$, let $\sqrt{\log _{2} x-1}=t$, then we have $\left\{\begin{array}{l}t-\frac{3}{2} t^{2}+\frac{1}{2}>0 \\ t \geqslant 0\end{array}\right.$, solving this gives $0 \leqslant t<1$, i.e., $0 \leqslant \log _{2} x-1<1$, $\therefore 2 \leqslant x<4$.