The expression \(3 \ln e^{3x}\) can be simplified using properties of logarithms and exponentials. Let's go through the simplification step-by-step.

### Step-by-Step Simplification

1. **Recall the property of logarithms**:  
   \(\ln(e^a) = a\) for any real number \(a\). This is because the natural logarithm function \(\ln(x)\) is the inverse of the exponential function \(e^x\).

2. **Apply the property**:  
   Substitute \(a = 3x\) into the logarithm property:

   \[
   \ln(e^{3x}) = 3x.
   \]

3. **Multiply by the coefficient**:  
   Now multiply by the coefficient 3:

   \[
   3 \ln(e^{3x}) = 3 \cdot 3x = 9x.
   \]

### Final Answer
The simplified form of the expression \(3 \ln e^{3x}\) is:

\[
9x.
\]

Would you like more details or have any questions? 

Here are some related questions:
1. What are the properties of logarithms?
2. How do exponential and logarithmic functions relate to each other?
3. What is the derivative of the expression \(3 \ln e^{3x}\) with respect to \(x\)?
4. How do we solve equations involving natural logarithms?
5. How does the natural logarithm differ from logarithms with other bases?

**Tip:** Remember that logarithms and exponentials are inverse functions; they "cancel out" when applied in the correct order.

Understand how to simplify the logarithmic expression 3ln(e^3x) using the properties of logarithms and exponentials. This step-by-step solution explains key concepts and formulas in logarithms and exponential functions. Suitable for students in Grades 9-12.

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