Change of Base Formula Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Change of Base Formula.

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

Concept Recap

A formula for converting a logarithm from one base to another: \log_b x = \frac{\ln x}{\ln b} = \frac{\log x}{\log b}.

Your calculator only has \ln and \log_{10} buttons. The change-of-base formula lets you compute ANY logarithm using whichever base you have available. It works because all logarithms are proportional to each other—changing base just changes the scale factor.

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: Any logarithm can be expressed as a ratio of two logarithms in the same base. This unifies all logarithm bases into one framework.

Common stuck point: The argument x goes in the numerator, the base b goes in the denominator. A mnemonic: 'argument on top, base on bottom.'

Sense of Study hint: On your calculator, compute ln(x) / ln(b) to find log base b of x. Check with a simple case first: ln(8)/ln(2) should give 3.

Worked Examples

Example 1

easy
Evaluate \log_5(20) using the change of base formula and a calculator.

Solution

  1. 1
    The change of base formula: \log_b(a) = \frac{\ln(a)}{\ln(b)} (or equivalently \frac{\log(a)}{\log(b)}).
  2. 2
    Apply: \log_5(20) = \frac{\ln(20)}{\ln(5)}.
  3. 3
    Calculate: \frac{\ln(20)}{\ln(5)} = \frac{2.9957}{1.6094} \approx 1.861.

Answer

\log_5(20) \approx 1.861
The change of base formula converts a logarithm from any base to a quotient of logarithms in a base your calculator supports (usually base 10 or base e). This is essential because most calculators only have \log and \ln buttons.

Example 2

medium
Show that \log_4(8) = \frac{3}{2} using the change of base formula.

Practice Problems

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

Example 1

medium
Simplify \log_9(27) to an exact fraction.

Example 2

hard
Prove that \log_a(b) \cdot \log_b(c) = \log_a(c) using the change of base formula.
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Background Knowledge

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

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