7.251. $x^{2-\lg ^{2} x-\lg x^{2}}-\frac{1}{x}=0$.

7.251. $x^{2-\log ^{2} x-\log x^{2}}-\frac{1}{x}=0$.
## Solution.

Domain of definition: $x>0$.

Let's write the equation in the form $x^{2-\lg ^{2} x-2 \lg x}=x^{-1}$. Taking the logarithm of both sides of the equation with base 10, we get

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

from which $\lg x=0$ or $\lg ^{2} x+2 \lg x-3=0$. From the first equation, $x_{1}=10^{0}=1$. Solving the second equation as a quadratic equation in terms of $\lg x$, we get $\lg x=-3$ and $\lg x=1$, from which $x_{2}=10^{-3}=0.001, x_{3}=10$.

Answer: 0.001; 1; 10.