7.115. $\lg ^{2}(100 x)+\lg ^{2}(10 x)=14+\lg \frac{1}{x}$.
## Solution.

Domain: $x>0$.

Taking logarithms, we have

$$
(\lg 100+\lg x)^{2}+(\lg 10+\lg x)^{2}=14-\lg x, \quad 2 \lg ^{2} x+7 \lg x-9=0
$$

Solving this equation as a quadratic in $\lg x$, we get $(\lg x)_{1}=-\frac{9}{2}$ or $(\lg x)_{2}=1$, from which $x_{1}=10^{-9 / 2}, x_{2}=10$.

Answer: $\quad 10^{-9 / 2} ; 10$.