Solve the following equation in the set of real numbers:

$$
5 \log _{\frac{x}{9}} x+\log _{\frac{9}{x}} x^{3}+8 \log _{9 x^{2}} x^{2}=2
$$
Using the properties of logarithms, we can reduce the equation to a quadratic equation, whose roots are 3 and $\sqrt{3}$. It is important not to forget to verify the correctness of the roots. At the very least, it must be noted that since we performed identical transformations, no new roots have appeared and no roots have been lost.