7.202. $5 \log _{x / 9} x+\log _{9 / x} x^{3}+8 \log _{9 x^{2}} x^{2}=2$.
## Solution.

Domain of definition: $\left\{\begin{array}{l}x>0, \\ x \neq \frac{1}{9}, \\ x \neq \pm \frac{1}{3} .\end{array}\right.$

Switch to base 9. We have

$$
\begin{aligned}
& \frac{5 \log _{9} x}{\log _{9} \frac{x}{9}}+\frac{\log _{9} x^{3}}{\log _{9} \frac{9}{x}}+\frac{8 \log _{9} x^{2}}{\log _{9} 9 x^{2}}=2 \Leftrightarrow \\
& \Leftrightarrow \frac{5 \log _{9} x}{\log _{9} x-1}+\frac{3 \log _{9} x}{1-\log _{9} x}+\frac{16 \log _{9} x}{1+2 \log _{9} x}=2 \Leftrightarrow 8 \log _{9}^{2} x-6 \log _{9} x+1=0
\end{aligned}
$$

Solving this equation as a quadratic in $\log _{9} x$, we get $\left(\log _{9} x\right)_{1}=\frac{1}{4}$ or $\left(\log _{9} x\right)_{2}=\frac{1}{2}$, from which $x_{1}=\sqrt{3}, x_{2}=3$.

Answer: $\sqrt{3} ; 3$.