ĐK: $x>0.$
$\left|\frac{1+i\sqrt{7}}{4} -\log_{2}x\right|\leq 1\Leftrightarrow \left|\frac{1}{4} -\log_{2}x +\frac{\sqrt{7}}{4}i \right|\leq 1$
$\Leftrightarrow \left ( \frac{1}{4} -\log_{2}x \right )^2 + \left ( \frac{\sqrt{7}}{4} \right )^2 \le 1 \Leftrightarrow \left ( \frac{1}{4} -\log_{2}x \right )^2 \le \frac{9}{16} $
$\Leftrightarrow -\frac{3}{4} \le \frac{1}{4} -\log_{2}x \le \frac{3}{4} \Leftrightarrow -\frac{1}{2} \le \log_{2}x \le 1 \Leftrightarrow \frac{1}{\sqrt 2} \le x \le 2.$