Natural Logarithm Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
Simplify ln⁡(x3)−2ln⁡(x)+ln⁡(e)\ln(x^3) - 2\ln(x) + \ln(e).

Solution

  1. 1
    Apply the power rule: ln⁡(x3)=3ln⁡(x)\ln(x^3) = 3\ln(x).
  2. 2
    Substitute: 3ln⁡(x)−2ln⁡(x)+ln⁡(e)3\ln(x) - 2\ln(x) + \ln(e).
  3. 3
    =ln⁡(x)+1= \ln(x) + 1 since ln⁡(e)=1\ln(e) = 1.

Answer

ln⁡(x)+1\ln(x) + 1
Logarithm properties — power rule (ln⁡(an)=nln⁡a\ln(a^n) = n\ln a), product rule, and quotient rule — apply to natural logarithms just as they do to logarithms of any base. Also, ln⁡(e)=1\ln(e) = 1 because e1=ee^1 = e.

About Natural Logarithm

The logarithm with base e≈2.71828e \approx 2.71828: ln⁡x=log⁡ex\ln x = \log_e x. It is the inverse function of exe^x.

Learn more about Natural Logarithm →

More Natural Logarithm Examples

Example 1 easy

Evaluate [formula].

Example 3 medium

Solve [formula] for [formula].

Example 4 hard

Find the derivative of [formula] and determine where [formula] is increasing.

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