Natural Logarithm Math Example 1

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

Example 1

easy
Evaluate ln⁡(e5)\ln(e^5).

Solution

  1. 1
    Recall that ln⁡(x)=log⁡e(x)\ln(x) = \log_e(x), so ln⁡\ln and exe^x are inverse functions.
  2. 2
    By the inverse property: ln⁡(ea)=a\ln(e^a) = a for any real number aa.
  3. 3
    Therefore ln⁡(e5)=5\ln(e^5) = 5.

Answer

55
The natural logarithm ln⁡\ln is the inverse of the exponential function exe^x. This means ln⁡(ea)=a\ln(e^a) = a and eln⁡a=ae^{\ln a} = a. These inverse relationships are fundamental to working with exponential and logarithmic expressions.

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 2 medium

Simplify [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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