3. The solution set of the inequality $\sqrt{\log _{2} x-1}+\frac{1}{2} \log _{\frac{1}{2}} x^{3}+2>0$ is
A. $[2,3)$
B. $(2,3]$
C. $[2,4)$
D. $(2,4]$
$$
\begin{array}{l}
\sqrt{\log _{2} x-1}+\frac{1}{2} \log _{\frac{1}{2}} x^{3}+2=\sqrt{\log _{2} x-1}-\frac{3}{2}\left(\log _{2} x-1\right)+\frac{1}{2}>0 \\
\xrightarrow{t=\sqrt{\log _{2} x-1} \geqslant 0} t-\frac{3}{2} t^{2}+\frac{1}{2}>0 \Rightarrow 3 t^{2}-2 t-1=(3 t+1)(t-1)<0 \\
\Rightarrow 0 \leqslant t<1 \Rightarrow 0 \leqslant \log _{2} x-1<1 \Rightarrow 2 \leqslant x<4 \text {. Therefore, the answer is } C \text {. } \\
\end{array}
$$