7.165. $3 \cdot 4^{x}+\frac{1}{3} \cdot 9^{x+2}=6 \cdot 4^{x+1}-\frac{1}{2} \cdot 9^{x+1}$.
Solution.

From the condition we have

$3 \cdot 4^{x}+\frac{1}{3} \cdot 81 \cdot 9^{x}=6 \cdot 4 \cdot 4^{x}-\frac{1}{2} \cdot 9 \cdot 9^{x} \Rightarrow 3 \cdot 9^{x}=2 \cdot 4^{x} \Rightarrow\left(\frac{9}{4}\right)^{x}=\frac{2}{3}$,

$\left(\frac{3}{2}\right)^{2 x}=\left(\frac{3}{2}\right)^{-1}$, from which $x=-\frac{1}{2}$.

Answer: $-\frac{1}{2}$.