$7.32 \log _{x}\left(9 x^{2}\right) \cdot \log _{3}^{2} x=4$.
7.32 Here $x>0, x \neq 1$. After transformations, we bring the equation to the form

$\left(2 \log _{x} 3+2\right) \log _{3}^{2} x=4$,

from which, considering that $\log _{x} 3 \cdot \log _{3} x=1$, we get

$2 \log _{3} x+2 \log _{3}^{2} x=4$.

Solving this equation using the substitution $\log _{3} x=y$, we find $x_{1}=\frac{1}{9}, x_{2}=3$.

Answer: $x_{1}=\frac{1}{9}, x_{2}=3$.