$7.22 \quad \lg ^{2}(100 x)+\lg ^{2}(10 x)=14+\lg \frac{1}{x}$.
7.22 Here $x>0$. After some transformations, we obtain the equivalent equation

$$
(2+\lg x)^{2}+(1+\lg x)^{2}=14-\lg x
$$

Letting $\lg x=y$, we arrive at the quadratic equation $2 y^{2}+7 y-9=0$, which has roots $y_{1}=-9 / 2, y_{2}=1$. Therefore, $x_{1}=10^{-9 / 2}, x_{2}=10$.

Answer: $x_{1}=10^{-9 / 2}, x_{2}=10$.