To solve the expression involving the natural logarithm and exponentiation, \( \ln(e^3) \), let's break it down:

### Step-by-Step Explanation:
1. **Recall the relationship between logarithms and exponents**:
   The natural logarithm \( \ln(x) \) is the inverse of the exponential function \( e^x \). Specifically, \( \ln(e^x) = x \).

2. **Apply this property**:
   In the given expression \( \ln(e^3) \), the argument of the logarithm is \( e^3 \). By the property mentioned, we can simplify \( \ln(e^3) \) to:

   \[
   \ln(e^3) = 3
   \]

### Final Answer:
\[
\ln(e^3) = 3
\]

Let me know if you need more details or further explanation!

Here are five related questions:
1. What is \( \ln(e^5) \)?
2. How would you simplify \( \ln(e^{x+2}) \)?
3. What is the value of \( \ln(1) \)?
4. Can you explain how \( e^{\ln(x)} \) simplifies for any \( x \)?
5. How does the base of the logarithm affect the simplification of \( \ln(e^x) \)?

**Tip**: Remember, the natural logarithm \( \ln(x) \) is shorthand for logarithm with base \( e \), where \( e \approx 2.718 \).

Learn how to simplify the natural logarithm ln(e^3) with a step-by-step explanation. This problem involves key mathematical concepts such as logarithmic and exponential functions, and simplifies the expression ln(e^3) using the inverse property of logarithms. Suitable for students in Grades 9-12.

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