1. **State the problem:** Solve the logarithmic equation $$\log_6(-4x) - \log_6 7 = \log_6 54$$ for $x$. 2. **Recall the logarithm subtraction rule:** $$\log_b A - \log_b B = \log_b \left(\frac{A}{B}\right)$$ provided $A, B > 0$. 3. **Apply the rule:** $$\log_6 \left(\frac{-4x}{7}\right) = \log_6 54$$ 4. **Since the logs are equal and the base is the same, their arguments must be equal:** $$\frac{-4x}{7} = 54$$ 5. **Solve for $x$:** $$-4x = 54 \times 7$$ $$-4x = 378$$ 6. **Divide both sides by $-4$:** $$x = \frac{378}{\cancel{-4}} \times \cancel{-1} = -\frac{378}{4}$$ 7. **Simplify the fraction:** $$x = -\frac{189}{2}$$ 8. **Check domain restrictions:** Argument of $\log_6(-4x)$ must be positive: $$-4x > 0 \implies x < 0$$ Our solution $x = -\frac{189}{2}$ is negative, so it is valid. **Final answer:** $$x = -\frac{189}{2}$$