7.082. $x^{\frac{\lg x+5}{3}}=10^{5+\lg x}$.
Solution.

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

Taking the logarithm of both sides of the equation with base 10, we have

$$
\lg x^{\frac{\lg x+5}{3}}=\lg 10^{5+\lg x}, \frac{\lg x+5}{3} \lg x=(5+\lg x) \lg 10, \lg ^{2} x+2 \lg x-15=0
$$

Solving this equation as a quadratic equation in terms of $\lg x$, we get $(\lg x)_{1}=-5$, or $(\lg x)_{2}=3$, from which $x_{1}=10^{-5}, x_{2}=1000$.

Answer: $10^{-5} ; 10^{3}$.