7.127. $\log _{2}\left(25^{x+3}-1\right)=2+\log _{2}\left(5^{x+3}+1\right)$
Solution.

Domain of definition: $25^{x+3}-1>0, 25^{x+3}>25^{0}, x>-3$.

From the condition

$\log _{2}\left(25^{3} \cdot 25^{x}-1\right)=\log _{2} 4\left(5^{3} \cdot 5^{x}+1\right) 25^{3} \cdot 5^{2 x}-1=4 \cdot 5^{3} \cdot 5^{x}+4$, $3125 \cdot 5^{2 x}-100 \cdot 5^{x}-1=0$,

from which, solving this equation as a quadratic in terms of $5^{x}$, we have $5^{x}=-\frac{1}{125}, \varnothing ;$ or $5^{x}=5^{-2}$, from which $x=-2$.

Answer: -2.

Solve the systems of equations (7.128-7.149):