jmc

Begin by combining logs: $${\log }_2\left(\frac{3x+9}{5x-3}\cdot \frac{5x-3}{x-2}\right)=2$$Notice that $5x-3$ cancels. We are left with: $${\log }_2\frac{3x+9}{x-2}=2$$Now, eliminate logs and solve: $$\begin{array}{rl}\frac{3x+9}{x-2}& =2^2\\ \Rightarrow 3x+9& =4(x-2)\\ \Rightarrow 3x+9& =4x-8\\ \Rightarrow \boxed{17}& =x\end{array}$$