7.117. $2^{\log _{3} x^{2}} \cdot 5^{\log _{3} x}=400$.
Solution.

Domain: $x>0$.

From the condition $4^{\log _{3} x} \cdot 5^{\log _{3} x}=400, 20^{\log _{3} x}=20^{2}$, hence $\log _{3} x=2$, $x=9$.

Answer: 9.