Natural Logarithm Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Natural Logarithm.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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

If e^x asks 'what do I get after growing continuously for time x?', then \ln x asks 'how long do I need to grow continuously to reach x?' The natural log measures time in the world of continuous growth.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: \ln x and e^x are perfect inverses: they undo each other. The natural log is 'natural' because \frac{d}{dx}\ln x = \frac{1}{x}—the simplest possible antiderivative of \frac{1}{x}.

Common stuck point: In math and science, \ln always means base e. But in computer science and some calculators, \log might mean base 2 or base 10. Always check the convention.

Sense of Study hint: Use the key inverse relationship: e^(ln x) = x and ln(e^x) = x. If stuck, convert to exponential form and solve from there.

Worked Examples

Example 1

easy
Evaluate \ln(e^5).

Solution

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

Answer

5
The natural logarithm \ln is the inverse of the exponential function e^x. This means \ln(e^a) = a and e^{\ln a} = a. These inverse relationships are fundamental to working with exponential and logarithmic expressions.

Example 2

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

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
Solve \ln(2x + 1) = 3 for x.

Example 2

hard
Find the derivative of f(x) = \ln(x^2 + 1) and determine where f is increasing.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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