1. **State the problem:** Solve the inequality $$\ln(x-2) + \ln\left(\frac{x}{3}\right) \le 0$$ for $x$. 2. **Use logarithm properties:** Recall that $$\ln(a) + \ln(b) = \ln(ab)$$ for $a>0$ and $b>0$. 3. **Combine the logarithms:** $$\ln\left((x-2) \cdot \frac{x}{3}\right) \le 0$$ 4. **Simplify inside the logarithm:** $$\ln\left(\frac{x(x-2)}{3}\right) \le 0$$ 5. **Rewrite inequality using exponential function:** Since $\ln(y) \le 0$ means $y \le 1$ for $y>0$, we have $$\frac{x(x-2)}{3} \le 1$$ 6. **Multiply both sides by 3:** $$x(x-2) \le 3$$ 7. **Expand the left side:** $$x^2 - 2x \le 3$$ 8. **Bring all terms to one side:** $$x^2 - 2x - 3 \le 0$$ 9. **Factor the quadratic:** $$x^2 - 2x - 3 = (x-3)(x+1)$$ 10. **Solve the inequality:** $$(x-3)(x+1) \le 0$$ This holds when $x$ is between the roots $-1$ and $3$, inclusive. 11. **Check domain restrictions:** - From $\ln(x-2)$, we need $x-2 > 0 \Rightarrow x > 2$. - From $\ln\left(\frac{x}{3}\right)$, we need $\frac{x}{3} > 0 \Rightarrow x > 0$. Combining domain restrictions: $x > 2$. 12. **Combine domain and solution:** The solution interval from the inequality is $[-1,3]$, but domain restricts to $x > 2$, so the final solution is $$2 < x \le 3$$ **Final answer:** $$\boxed{2 < x \le 3}$$