7.022. $\sqrt{\log _{3} x^{9}}-4 \log _{9} \sqrt{3 x}=1$.
Solution.

Domain of definition: $\left\{\begin{array}{l}x>0, \\ \log _{3} x>0, x>1\end{array}\right.$

$$
\begin{aligned}
& \sqrt{\log _{3} x^{9}}=1+4 \log _{9} \sqrt{3 x} \Leftrightarrow \sqrt{9 \log _{3} x}=1+\log _{3} 3 x \Leftrightarrow \\
& \Leftrightarrow \sqrt{9 \log _{3} x}=1+\log _{3} 3+\log _{3} x \Leftrightarrow \sqrt{9 \log _{3} x}=2+\log _{3} x
\end{aligned}
$$

Squaring both sides of the equation, we get

$$
9 \log _{3} x=4+4 \log _{3} x+\log _{3}^{2} x \Leftrightarrow \log _{3}^{2} x-5 \log _{3} x+4=0
$$

Solving this equation as a quadratic equation in terms of $\log _{3} x$, we have $\left(\log _{3} x\right)_{1}=1,\left(\log _{3} x\right)_{2}=4$, from which $x_{1}=3, x_{2}=3^{4}=81$.

Answer: $3 ; 81$.