1. **State the problem:** Solve the logarithmic equation $$3 \log_2 - 2 \log_2 \frac{x}{3} = 2 \log_2 3 + 1$$. 2. **Rewrite the equation clearly:** Note that $$\log_2$$ without an argument is incomplete, so we assume it means $$\log_2 2$$ which equals 1. The equation becomes: $$3 \cdot 1 - 2 \log_2 \frac{x}{3} = 2 \log_2 3 + 1$$ 3. **Simplify the left side:** $$3 - 2 \log_2 \frac{x}{3} = 2 \log_2 3 + 1$$ 4. **Isolate the logarithmic term:** $$-2 \log_2 \frac{x}{3} = 2 \log_2 3 + 1 - 3$$ $$-2 \log_2 \frac{x}{3} = 2 \log_2 3 - 2$$ 5. **Divide both sides by -2:** $$\log_2 \frac{x}{3} = \cancel{-\frac{2}{2}} \log_2 3 + \cancel{-\frac{2}{2}}$$ $$\log_2 \frac{x}{3} = - \log_2 3 + 1$$ 6. **Rewrite 1 as $$\log_2 2$$:** $$\log_2 \frac{x}{3} = - \log_2 3 + \log_2 2$$ 7. **Use logarithm properties:** $$\log_2 \frac{x}{3} = \log_2 \frac{2}{3}$$ 8. **Since logarithms are equal, their arguments are equal:** $$\frac{x}{3} = \frac{2}{3}$$ 9. **Solve for $$x$$:** $$x = 2$$ **Final answer:** $$x = 2$$