$7.13 x^{\frac{\lg x+5}{3}}=10^{\lg x+1}$.
7.13 Since the logarithmic function is defined for $x>0$, both the left and right sides of the given equation are positive. Taking their logarithms to base 10 and using formulas (7.6) and (7.2), we arrive at the equation

$\frac{\lg x+5}{3} \lg x=\lg x+1$.

Let's make the substitution $y=\lg x$ and solve the equation $y^{2}+5 y=3 y+3$. We have $y^{2}+2 y-3=0$, from which $y_{1}=-3, \quad y_{2}=1$. From the equation $\lg x=-3$ we get $x_{1}=10^{-3}$, and from the equation $\lg x=1$ we find $x_{2}=10$.

Answer: $x_{1}=0.001, x_{2}=10$.