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The exercise asks to express certain values in terms of the natural logarithm, using the relationship between exponentials and logarithms. Here’s how to approach each part:

1. **Understanding the natural logarithm (ln)**: The natural logarithm \( \ln(x) \) is the inverse of the exponential function \( e^x \). For example, if \( e^3 \approx 20.1 \), then \( \ln(20.1) \approx 3 \).

Let's go through each part:

- **(a) \( e^1 \approx 2.718 \)**
  - Since \( e^1 \) represents the natural base \( e \), it’s simply the number \( e \) itself. No further transformation with logarithms is needed here.

- **(b) \( e^0 = 1 \)**
  - Since any number to the power of zero is 1, this is straightforward.

- **(c) \( e^{-1} \approx 0.37 \)**
  - Here, \( e^{-1} \) is the reciprocal of \( e \), or \( \frac{1}{e} \).

- **(d) \( 0.82 \approx e^{-0.2} \)**
  - This means that \( \ln(0.82) \approx -0.2 \), using the logarithmic relationship \( e^x = y \Rightarrow x = \ln(y) \).

- **(e) \( e^{3/4} \approx 2.12 \)**
  - For this, we note that \( \ln(2.12) \approx \frac{3}{4} \).

These are examples of how the natural logarithm can represent exponential expressions.

Would you like more detailed steps on any of these parts? 

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### Expanding Questions:
1. How does the natural logarithm help in solving exponential equations?
2. Why is \( e \) considered a special number in mathematics?
3. What are some real-world applications of exponential and logarithmic functions?
4. Can you explain the significance of the base \( e \) in calculus?
5. How would these principles apply if the base was not \( e \) but another number?

### Tip:
Remember, \( \ln(e^x) = x \) simplifies many calculations involving exponentials and logarithms.

Schreiben Sie mithilfe des natürlichen Logarithmus.

Learn how to express exponential values like e^1 ≈ 2.718, e^0 = 1, and e^(-1) ≈ 0.37 using natural logarithms. This exercise covers concepts of natural logarithms and their relationship with exponential functions, suitable for students in Grades 10-12.

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