China Mathematical Competition (Hainan)

$$\{\begin{array}{l}\sqrt{{\log }_{2}x-1}-\frac{3}{2}{\log }_{2}x+\frac{3}{2}+\frac{1}{2}>0,\\ {\log }_{2}x-1\ge 0.\end{array}$$ Let $t=\sqrt{{\log }_{2}x-1}$, we have $$\{\begin{array}{l}t-\frac{3}{2}{t}^2+\frac{1}{2}>0,\\ t\ge 0.\end{array}$$

The solution of the above inequalities is $0\le t<1$, or $0\le {\log }_{2}x-1<1$, which implies that $2\le x<4$. Answer: C.