7.234. $\frac{10 x^{2 \lg x}}{x^{3}}=\frac{x^{3 \lg x}}{10}$.
## Solution.

Domain of definition: $0<x \neq 1$.

From the condition, we have $\frac{x^{2 \lg ^{2} x}}{x^{3} \cdot x^{3 \lg x}}=\frac{1}{100} \Leftrightarrow x^{2 \lg ^{2} x-3 \lg x-3}=10^{-2}$. Taking the logarithm of both sides of this equation with base 10, we get

$$
\begin{aligned}
& \lg x^{2 \lg ^{2} x-3 \lg x-3}=\lg 10^{-2} \Leftrightarrow\left(2 \lg ^{2} x-3 \lg x-3\right) \lg x=-2 \Leftrightarrow \\
& \Leftrightarrow 2 \lg ^{3} x-3 \lg ^{2} x-3 \lg x+2=0 \Leftrightarrow 2\left(\lg ^{3} x+1\right)-3 \lg x(\lg x+1)=0 \Leftrightarrow \\
& \Leftrightarrow 2(\lg x+1)\left(\lg ^{2} x-\lg x+1\right)-3 \lg x(\lg x+1)=0 \Leftrightarrow \\
& \Leftrightarrow(\lg x+1)\left(2 \lg ^{2} x-5 \lg x+2\right)=0 \Rightarrow \\
& \Rightarrow \lg x+1=0 \text { or } 2 \lg ^{2} x-5 \lg x+2=0
\end{aligned}
$$

From the first equation, we have $\lg x=-1, x_{1}=\frac{1}{10}$, and from the second $\lg x=\frac{1}{2}$, $x_{2}=\sqrt{10}$ or $\lg x=2, x_{3}=100$.

Answer: 0,$1 ; \sqrt{10} ; 100$.