7.091. $\log _{2}\left(4 \cdot 3^{x}-6\right)-\log _{2}\left(9^{x}-6\right)=1$.
## Solution.

oDZ: $\left\{\begin{array}{l}4 \cdot 3^{x}-6>0 \\ 9^{x}-6>0\end{array}\right.$

We have $\log _{2} \frac{4 \cdot 3^{x}-6}{3^{2 x}-6}=1, \frac{4 \cdot 3^{x}-6}{3^{2 x}-6}=2 \Rightarrow 3^{2 x}-2 \cdot 3^{x}-3=0$. Solving it as a quadratic equation in terms of $3^{x}$, we find $3^{x}=-1, \varnothing$; or $3^{x}=3$, from which $x=1$.

Answer: 1.