7.113. $x^{\lg x}=1000 x^{2}$.
Solution.

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

Taking the logarithm of both sides of the equation with base 10, we get $\lg x^{\lg x}=\lg 1000 x^{2}, \lg x \lg x=\lg 1000+\lg x^{2}, \lg ^{2} x-2 \lg x-3=0$. Solving this equation as a quadratic equation in terms of $\lg x$, we get $(\lg x)_{1}=-1$ or $(\lg x)_{2}=3$, from which $x_{1}=0.1, x_{2}=1000$.

Answer: 0.1; 1000.