1. **State the problem:** Solve the equation $9\ln(x+1) = -27$ for $x$. 2. **Recall the formula and rules:** The natural logarithm function $\ln(y)$ is the inverse of the exponential function $e^y$. To solve for $x$, we will isolate $\ln(x+1)$ and then exponentiate both sides to remove the logarithm. 3. **Isolate the logarithm:** $$9\ln(x+1) = -27$$ Divide both sides by 9: $$\cancel{9}\ln(x+1) = \frac{-27}{\cancel{9}}$$ $$\ln(x+1) = -3$$ 4. **Exponentiate both sides:** Apply $e^{(\cdot)}$ to both sides to cancel the logarithm: $$e^{\ln(x+1)} = e^{-3}$$ Since $e^{\ln(y)} = y$, we get: $$x+1 = e^{-3}$$ 5. **Solve for $x$:** $$x = e^{-3} - 1$$ 6. **Final answer:** $$\boxed{x = e^{-3} - 1}$$ This is the exact solution. Numerically, $e^{-3} \approx 0.0498$, so $x \approx -0.9502$. This completes the solution.