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.