7.256. $\left(1+\frac{x}{2}\right) \log _{2} 3-\log _{2}\left(3^{x}-13\right)=2$.
Solution.

Domain: $3^{x}-13>0 \Leftrightarrow x>\log _{3} 13$.

From the condition we have

$\log _{2} 3^{1+\frac{x}{2}}-\log _{2}\left(3^{x}-13\right)=2 \Leftrightarrow \log _{2} \frac{3^{1+\frac{x}{2}}}{3^{x}-13}=2 \Leftrightarrow \frac{3^{1+\frac{x}{2}}}{3^{x}-13}=4 \Leftrightarrow$

$\Leftrightarrow 4 \cdot 3^{x}-3 \cdot 3^{\frac{x}{2}}-52=0$.

Solving this equation as a quadratic in terms of $3^{\frac{x}{2}}$, we get $3^{\frac{x}{2}}=-\frac{13}{4}$ (no solutions), or $3^{\frac{x}{2}}=4 \Rightarrow x=4 \log _{3} 2$.

Answer: $4 \log _{3} 2$.