1. **State the problem:** Solve the equation $$\ln(x - 9) + \ln 2 = \ln 45$$ for $x$. 2. **Recall the logarithm property:** The sum of logarithms with the same base can be combined as the logarithm of the product: $$\ln a + \ln b = \ln(ab)$$ 3. **Apply the property:** $$\ln(x - 9) + \ln 2 = \ln\big((x - 9) \times 2\big) = \ln(2x - 18)$$ 4. **Rewrite the equation:** $$\ln(2x - 18) = \ln 45$$ 5. **Since the natural logarithm function is one-to-one, set the arguments equal:** $$2x - 18 = 45$$ 6. **Solve for $x$:** $$2x = 45 + 18$$ $$2x = 63$$ $$x = \frac{63}{2}$$ 7. **Check the domain:** The argument of the logarithm must be positive: $$x - 9 > 0 \implies x > 9$$ Since $x = 31.5$ is greater than 9, it is valid. **Final answer:** $$x = 31.5$$