Practice Change of Base Formula in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Quick 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.

Example 1

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

Example 2

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

Example 3

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

Example 4

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