1. **State the problem:** Solve the equation $-6 \log_{10} (3(x-3)) = -24$ for $x$. 2. **Isolate the logarithm:** Divide both sides by $-6$ to isolate the logarithmic expression. $$-6 \log_{10} (3(x-3)) = -24$$ $$\cancel{-6} \log_{10} (3(x-3)) = \cancel{-6} \times 4$$ $$\log_{10} (3(x-3)) = 4$$ 3. **Rewrite the logarithmic equation in exponential form:** Recall that $\log_b a = c$ means $a = b^c$. $$3(x-3) = 10^4$$ 4. **Simplify the exponential:** $$3(x-3) = 10000$$ 5. **Solve for $x$:** $$x-3 = \frac{10000}{3}$$ $$x = 3 + \frac{10000}{3} = \frac{9}{3} + \frac{10000}{3} = \frac{10009}{3}$$ 6. **Check the domain:** The argument of the logarithm must be positive: $$3(x-3) > 0 \implies x-3 > 0 \implies x > 3$$ Since $\frac{10009}{3} \approx 3336.33 > 3$, the solution is valid. **Final answer:** $$x = \frac{10009}{3}$$