Change of Base Formula Math Example 2

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

Example 2

medium
Show that log⁡4(8)=32\log_4(8) = \frac{3}{2} using the change of base formula.

Solution

  1. 1
    Apply change of base with base 2: log⁡4(8)=log⁡2(8)log⁡2(4)\log_4(8) = \frac{\log_2(8)}{\log_2(4)}.
  2. 2
    log⁡2(8)=3\log_2(8) = 3 since 23=82^3 = 8, and log⁡2(4)=2\log_2(4) = 2 since 22=42^2 = 4.
  3. 3
    Therefore log⁡4(8)=32\log_4(8) = \frac{3}{2}.

Answer

log⁡4(8)=32\log_4(8) = \frac{3}{2}
When changing base, you can choose any convenient base — not just 10 or ee. Here, base 2 is ideal because both 4 and 8 are powers of 2, making the computation exact without a calculator.

About Change of Base Formula

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

Learn more about Change of Base Formula →

More Change of Base Formula Examples

Example 1 easy

Evaluate [formula] using the change of base formula and a calculator.

Example 3 medium

Simplify [formula] to an exact fraction.

Example 4 hard

Prove that [formula] using the change of base formula.

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