7.7 $\quad 2^{\log _{3} x^{2}} \cdot 5^{\log _{3} x}=400$.
7.7 The logarithmic function $y=\log _{3} x$ is defined for $x>0$. Therefore, according to formula (7.6), we have $\log _{3} x^{2}=2 \log _{3} x$. Consequently, $2^{2 \log _{3} x} \cdot 5^{\log _{3} x}=20^{2} ; 4^{\log _{3} x} \cdot 5^{\log _{3} x}=20^{2} ; 20^{\log _{3} x}=20^{2}$. From this, $\log _{3} x=2$, i.e., $x=3^{2}=9$.

Answer: $x=9$.