$$\begin{cases} \sqrt{\log_2 x - 1} - \frac{3}{2} \log_2 x + \frac{3}{2} + \frac{1}{2} > 0, \\ \log_2 x - 1 \ge 0. \end{cases}$$
Let $t = \sqrt{\log_2 x - 1}$, we have
$$\begin{cases} t - \frac{3}{2}t^2 + \frac{1}{2} > 0, \\ t \ge 0. \end{cases}$$

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.