7.175. $\lg \left(x^{3}+8\right)-0.5 \lg \left(x^{2}+4 x+4\right)=\lg 7$.
## Solution.

Domain of definition: $x+2>0, x>-2$.

Rewrite the equation as

$$
\begin{aligned}
& \lg (x+2)\left(x^{2}-2 x+4\right)-0.5 \lg (x+2)^{2}=\lg 7 \Leftrightarrow \\
& \Leftrightarrow \lg (x+2)\left(x^{2}-2 x+4\right)-\lg (x+2)=\lg 7 \Leftrightarrow \lg \frac{(x+2)\left(x^{2}-2 x+4\right)}{x+2}=\lg 7 \Leftrightarrow \\
& \Leftrightarrow x^{2}-2 x+4=7, x^{2}-2 x-3=0, \text { from which } x_{1}=-1, x_{2}=3
\end{aligned}
$$

Answer: $\quad-1 ; 3$.