Math Problem Statement

Schreiben Sie mithilfe des natürlichen Logarithmus.

Solution

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)\ln(x) is the inverse of the exponential function exe^x. For example, if e320.1e^3 \approx 20.1, then ln(20.1)3\ln(20.1) \approx 3.

Let's go through each part:

  • (a) e12.718e^1 \approx 2.718

    • Since e1e^1 represents the natural base ee, it’s simply the number ee itself. No further transformation with logarithms is needed here.
  • (b) e0=1e^0 = 1

    • Since any number to the power of zero is 1, this is straightforward.
  • (c) e10.37e^{-1} \approx 0.37

    • Here, e1e^{-1} is the reciprocal of ee, or 1e\frac{1}{e}.
  • (d) 0.82e0.20.82 \approx e^{-0.2}

    • This means that ln(0.82)0.2\ln(0.82) \approx -0.2, using the logarithmic relationship ex=yx=ln(y)e^x = y \Rightarrow x = \ln(y).
  • (e) e3/42.12e^{3/4} \approx 2.12

    • For this, we note that ln(2.12)34\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?


Expanding Questions:

  1. How does the natural logarithm help in solving exponential equations?
  2. Why is ee 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 ee in calculus?
  5. How would these principles apply if the base was not ee but another number?

Tip:

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

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Math Problem Analysis

Mathematical Concepts

Exponential Functions
Natural Logarithms

Formulas

e^x ≈ value
ln(value) = x where e^x = value

Theorems

Inverse Property of Exponential and Logarithmic Functions

Suitable Grade Level

Grades 10-12