Solve the exponential equation: 5^(2(x+3)) = 7^(x+3)

We have two possibilities:
Possibility 1: $x+3 = 0$

$$x = -3$$

Possibility 2: $2\ln(5) - \ln(7) = 0$
This simplifies to $\ln(5^2) = \ln(7)$, which means $\ln(25) = \ln(7)$. Since $25 \neq 7$, this statement is false, and this factor cannot be zero.

Therefore, the only valid solution is from Possibility 1.