9.120. Find the integer values of $x$ that satisfy the inequality $\log _{4} x+\log _{2}(\sqrt{x}-1)<\log _{2} \log _{\sqrt{5}} 5$.
## Solution.

Let's switch to base 2. We have $\frac{1}{2} \log _{2} x+\log _{2}(\sqrt{x}-1) 0 , } \\
{ \sqrt { x - 1 } > 0 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
(\sqrt{x})^{2}-\sqrt{x}-20, \\
\sqrt{x}>1
\end{array}\right.\right. \\
& \Leftrightarrow\left\{\begin{array}{l}
-11
\end{array} \Leftrightarrow\right. \\
& \Leftrightarrow\left\{\begin{array}{l}
0 \leq x1
\end{array} \Leftrightarrow 1<x<4 .\right.
\end{aligned}
$$

Answer: $x_{1}=2 ; x_{2}=3$.