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 \(x > 2\).

Learn how to solve the equation 3ln(x - 2) = 9 step by step. This problem explores the use of logarithmic properties, exponential functions, and algebraic manipulation, suitable for students in Grades 10-12.

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