7.051. $\frac{\lg 8-\lg (x-5)}{\lg \sqrt{x+7}-\lg 2}=-1$.
## Solution.

Domain of definition: $\left\{\begin{array}{l}x-5>0, \\ x+7>0, \quad x>5 . \\ \sqrt{x+7} \neq 2,\end{array}\right.$

## From the condition

$$
\begin{aligned}
& \lg 8-\lg (x-5)=\lg 2-\lg \sqrt{x+7}, \quad \lg \frac{8}{x-5}=\lg \frac{2}{\sqrt{x+7}} \\
& \frac{8}{x-5}=\frac{2}{\sqrt{x+7}}, \quad 4 \sqrt{x+7}=x-5, \quad 16 x+112=x^{2}-10 x+25, x>5
\end{aligned}
$$

We have $x^{2}-26 x-87=0$, from which $x_{1}=29, x_{2}=-3 ; x_{2}=-3$ does not satisfy the domain of definition.

Answer: 29.