5. (2004 National High School Mathematics Competition Question) The solution set of the inequality $\sqrt{\log _{2} x-1}+\frac{1}{2} \log _{4} x^{3}+2>0$ is
A. $[2,3]$
B. $(2,3)$
C. $[2,4)$
D. $(2,4)$
5. Solution 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, we get $0 \leqslant t<1$. That is, $0 \leqslant \log _{2} x-1<1$, so $2 \leqslant x<4$. Therefore, the answer is $\mathrm{C}$.