7.028. $5^{2\left(\log _{5} 2+x\right)}-2=5^{x+\log _{5} 2}$.
## Solution.

$\left(5^{x+\log _{5} 2}\right)^{2}-5^{x+\log _{5} 2}-2=0$; solving this equation as a quadratic equation in terms of $5^{x+\log _{5} 2}$, we find $5^{x+\log _{5} 2}=-1$ and $5^{x+\log _{5} 2}=2 ; 5^{x+\log _{5} 2}=-1$ has no solutions.

Thus,

$$
5^{x+\log _{5} 2}=2 \Rightarrow \log _{5} 5^{x+\log _{5} 2}=\log _{5} 2, x+\log _{5} 2=\log _{5} 2
$$

from which $x=0$.

Answer: 0.