7.040. $\lg \left(x^{2}+1\right)=2 \lg ^{-1}\left(x^{2}+1\right)-1$.

7.040. $\lg \left(x^{2}+1\right)=2 \lg ^{-1}\left(x^{2}+1\right)-1$.
## Solution.

Domain of definition: $x \neq 0$.

$$
\lg \left(x^{2}+1\right)=\frac{2}{\lg \left(x^{2}+1\right)}-1, \lg ^{2}\left(x^{2}+1\right)+\lg \left(x^{2}+1\right)-2=0
$$

Solving this equation as a quadratic equation in terms of $\lg \left(x^{2}+1\right)$, we find $\lg \left(x^{2}+1\right)=-2$ and $\lg \left(x^{2}+1\right)=1$. From this, $x^{2}+1=0.01, x^{2}=-0.99, \varnothing$. $x^{2}+1=10, x^{2}=9$. Then $x_{1,2}= \pm 3$.

Answer: $-3 ; 3$.