We are solving the equation $3 \ln(x - 2) = 9$. Let’s go step by step.

Step 1: Isolate the logarithmic term

$\ln(x - 2) = \frac{9}{3} = 3$

Step 2: Rewrite the equation in exponential form

Recall that $\ln(y) = a$ is equivalent to $y = e^a$.
Thus: $x - 2 = e^3$

Step 3: Solve for $x$

$x = e^3 + 2$

Step 4: Approximate the value (optional)

$e^3 \approx 20.0855$ $x \approx 20.0855 + 2 = 22.0855$

Final Answer

$x = e^3 + 2 \quad \text{or approximately} \quad x \approx 22.0855$

Would you like further details on the steps or an explanation of logarithmic and exponential relationships?

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Here are 5 related questions to extend your understanding:

1. How do you solve equations with natural logarithms and multiple $x$-terms?
2. What are the properties of logarithms, such as the product, quotient, and power rules?
3. How does the natural logarithm compare to the common logarithm ($\log_{10}$)?
4. How can you solve equations involving $\ln$ on both sides?
5. What is the derivative of $\ln(x)$ in calculus?

Tip: Always check the domain when solving logarithmic equations. In this case, $x - 2 > 0$ implies $x > 2$.