9.137. $\left(\log _{2} x\right)^{4}-\left(\log _{\frac{1}{2}} \frac{x^{3}}{8}\right)^{2}+9 \cdot \log _{2} \frac{32}{x^{2}}<4 \cdot\left(\log _{\frac{1}{2}} x\right)^{2}$.
## Solution.

Domain of definition: $x>0$.

Let's switch to base 2. We have

$$
\begin{aligned}
& \log _{2}^{4} x-\left(\log _{2} \frac{x^{3}}{8}\right)^{2}+9 \log _{2} \frac{32}{x^{2}}-4 \log _{2}^{\frac{2}{2}} x 2 } \\
{ \operatorname { log } _ { 2 } x 4 \\
0<x<\frac{1}{4}
\end{aligned}\right.} \\
\text { 0 }
\end{aligned}\right.\right.
$$

Using the number line, we get $x \in\left(\frac{1}{8} ; \frac{1}{4}\right) \cup(4 ; 8)$.

![](https://cdn.mathpix.com/cropped/2024_05_21_3edb7be591bff54ce350g-0664.jpg?height=173&width=504&top_left_y=194&top_left_x=289)

Answer: $x \in\left(\frac{1}{8} ; \frac{1}{4}\right) \cup(4 ; 8)$