1. **State the problem:** Solve the logarithmic equation $$\log_5 (x - 18) - \log_5 x = \log_5 7$$ for $x$. 2. **Recall the logarithm subtraction rule:** For logarithms with the same base, $$\log_b A - \log_b B = \log_b \left(\frac{A}{B}\right)$$. 3. **Apply the rule to the left side:** $$\log_5 (x - 18) - \log_5 x = \log_5 \left(\frac{x - 18}{x}\right)$$ 4. **Rewrite the equation:** $$\log_5 \left(\frac{x - 18}{x}\right) = \log_5 7$$ 5. **Since the logs are equal and the base is the same, set the arguments equal:** $$\frac{x - 18}{x} = 7$$ 6. **Solve the equation:** $$\frac{x - 18}{x} = 7$$ Multiply both sides by $x$: $$\cancel{x} \cdot \frac{x - 18}{\cancel{x}} = 7x$$ $$x - 18 = 7x$$ 7. **Isolate $x$:** $$x - 7x = 18$$ $$-6x = 18$$ 8. **Divide both sides by $-6$:** $$\frac{-6x}{\cancel{-6}} = \frac{18}{\cancel{-6}}$$ $$x = -3$$ 9. **Check the domain:** The arguments of the logarithms must be positive: - $x - 18 > 0 \Rightarrow x > 18$ - $x > 0$ Since $x = -3$ does not satisfy $x > 18$, it is not a valid solution. 10. **Conclusion:** There is no solution to the equation in the domain of the logarithm functions. **Final answer:** No solution.