1. **State the problem:** Solve the equation $$-6 \log_{3}(x-3) = -24$$ for $x$. 2. **Isolate the logarithm:** Divide both sides by $-6$ to isolate the logarithmic term. $$\cancel{-6} \log_{3}(x-3) = \cancel{-6} \times 4$$ which simplifies to $$\log_{3}(x-3) = 4$$ 3. **Rewrite the logarithmic equation in exponential form:** Recall that $\log_{a}(b) = c$ means $a^{c} = b$. So, $$3^{4} = x - 3$$ 4. **Calculate the power:** $$3^{4} = 3 \times 3 \times 3 \times 3 = 81$$ So, $$81 = x - 3$$ 5. **Solve for $x$:** Add 3 to both sides. $$x = 81 + 3 = 84$$ 6. **Check the domain:** The argument of the logarithm must be positive: $$x - 3 > 0 \Rightarrow x > 3$$ Since $84 > 3$, the solution is valid. **Final answer:** $$x = 84$$