1. **State the problem:** Solve the equation $\ln 2 - \ln (x + 1) = 3$ 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:** $$\ln 2 - \ln (x + 1) = \ln \left(\frac{2}{x+1}\right)$$ 4. **Rewrite the equation:** $$\ln \left(\frac{2}{x+1}\right) = 3$$ 5. **Exponentiate both sides to remove the logarithm:** $$e^{\ln \left(\frac{2}{x+1}\right)} = e^3$$ Since $e^{\ln y} = y$, we get: $$\frac{2}{x+1} = e^3$$ 6. **Solve for $x$:** Multiply both sides by $x+1$: $$2 = e^3 (x+1)$$ Divide both sides by $e^3$: $$\frac{2}{e^3} = x + 1$$ Subtract 1 from both sides: $$x = \frac{2}{e^3} - 1$$ 7. **Final answer:** $$\boxed{x = \frac{2}{e^3} - 1}$$ This is the solution for $x$ in terms of the exponential constant $e$.