7.041. $\lg \sqrt{5^{x(13-x)}}+11 \lg 2=11$.
Solution.

$\lg 5^{\frac{x(13-x)}{2}}+\lg 2^{11}=11, \lg \left(5^{\frac{x(13-x)}{2}} \cdot 2^{11}\right)=11$

From here we have $5^{\frac{x(13-x)}{2}} \cdot 2^{11}=10^{11}, 5^{\frac{x(13-x)}{2}}=5^{11}$. Then $\frac{x(13-x)}{2}=11$, $x^{2}-13 x+22=0$, from which $x_{1}=2 ; x_{2}=11$.

Answer: $2 ; 11$.