simplify each expression ln e3 ln e2y

Daniel Parker

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01-03-2025

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Understanding how to simplify logarithmic expressions is crucial in mathematics and many scientific fields. This article focuses on simplifying two specific expressions involving the natural logarithm (ln): ln(e³) and ln(e²ʸ). We'll break down the process step-by-step, explaining the underlying principles along the way. By the end, you'll be able to confidently tackle similar problems.

Understanding Natural Logarithms

The natural logarithm, denoted as ln(x) or logₑ(x), is the logarithm to the base e, where e is Euler's number, approximately equal to 2.71828. It's the inverse function of the exponential function eˣ. This means that if ln(a) = b, then eᵇ = a.

This inverse relationship is key to simplifying expressions like ln(e³).

Simplifying ln(e³)

The expression ln(e³) asks: "To what power must e be raised to equal e³?" The answer is clearly 3. Therefore:

ln(e³) = 3

This is a direct application of the inverse relationship between the natural logarithm and the exponential function. The natural logarithm and the exponential function with base e cancel each other out, leaving only the exponent.

Simplifying ln(e²ʸ)

The expression ln(e²ʸ) follows the same principle. We're asking: "To what power must e be raised to equal e²ʸ?" The answer, again, is the exponent: 2y. Therefore:

ln(e²ʸ) = 2y

This simplification works regardless of the value of y. The natural logarithm and the exponential function with base e cancel each other, leaving the exponent.

Key Principles and Further Applications

The core principle demonstrated in these examples is the inverse relationship between the natural logarithm and the exponential function with base e. This principle extends to more complex expressions. For example:

• ln(eˣ⁺ʸ) = x + y
• ln(eˣ/eʸ) = x - y
• ln(eˣ * eʸ) = x + y

Remember that this simplification only applies when the base of the exponential function is e and the natural logarithm is used. Different logarithmic bases require different approaches.

Practice Problems

To solidify your understanding, try simplifying these expressions:

1. ln(e⁷)
2. ln(e⁻²)
3. ln(e^(x+2))

Understanding these fundamental simplifications will greatly enhance your ability to solve more complex problems in calculus, physics, and other fields that utilize logarithmic functions. Remember to always apply the inverse relationship between the natural logarithm and the exponential function with base e.