1. **State the problem:** Solve the logarithmic equation $$\ln(2x - 8) - \ln(x - 4) = \ln(x)$$ for $x$. 2. **Recall the logarithm property:** The difference of logarithms can be written as the logarithm of a quotient: $$\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$$ 3. **Apply the property:** $$\ln\left(\frac{2x - 8}{x - 4}\right) = \ln(x)$$ 4. **Since $\ln(A) = \ln(B)$ implies $A = B$ (for $A,B > 0$), set the arguments equal:** $$\frac{2x - 8}{x - 4} = x$$ 5. **Simplify numerator:** $$2x - 8 = 2(x - 4)$$ 6. **Rewrite the equation:** $$\frac{2(x - 4)}{x - 4} = x$$ 7. **Cancel common factor $x - 4$ (noting $x \neq 4$ to avoid division by zero):** $$\frac{2\cancel{(x - 4)}}{\cancel{(x - 4)}} = x \implies 2 = x$$ 8. **Check domain restrictions:** - Arguments of logarithms must be positive: - $2x - 8 > 0 \Rightarrow 2x > 8 \Rightarrow x > 4$ - $x - 4 > 0 \Rightarrow x > 4$ - $x > 0$ 9. **Check if solution $x=2$ satisfies domain:** - $2$ is not greater than $4$, so $x=2$ is not in the domain. 10. **Conclusion:** No valid solution exists because the only candidate $x=2$ is outside the domain. **Final answer:** No solution.