7.031. $\log _{5}(x-2)+\log _{\sqrt{5}}\left(x^{3}-2\right)+\log _{0.2}(x-2)=4$.
## Solution.

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

From the condition we have

$$
\log _{5}(x-2)+2 \log _{5}\left(x^{3}-2\right)-\log _{5}(x-2)=4, \log _{5}\left(x^{3}-2\right)=2
$$

from which $x^{3}-2=25, x^{3}=27$. Then $x=3$.

Answer: 3.