1. **State the problem:** Solve the equation $$\ln 8 - \ln (-2x) = \ln 13$$ for $x$. 2. **Recall the logarithm property:** The difference of logarithms is the logarithm of the quotient: $$\ln a - \ln b = \ln \left(\frac{a}{b}\right)$$ 3. **Apply the property:** $$\ln 8 - \ln (-2x) = \ln \left(\frac{8}{-2x}\right) = \ln 13$$ 4. **Set the arguments equal:** Since $\ln A = \ln B$ implies $A = B$ (for $A,B>0$), we have: $$\frac{8}{-2x} = 13$$ 5. **Solve for $x$:** Multiply both sides by $-2x$: $$8 = 13 \times (-2x)$$ $$8 = -26x$$ Divide both sides by $-26$: $$x = \frac{8}{-26} = -\frac{4}{13}$$ 6. **Check domain restrictions:** The argument of the logarithm must be positive: $$-2x > 0 \implies x < 0$$ Our solution $x = -\frac{4}{13}$ satisfies this. **Final answer:** $$x = -\frac{4}{13}$$