Change of Base Formula Math Example 3

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

Example 3

medium
Simplify log⁡9(27)\log_9(27) to an exact fraction.

Solution

  1. 1
    Change to base 3: log⁡9(27)=log⁡3(27)log⁡3(9)=32\log_9(27) = \frac{\log_3(27)}{\log_3(9)} = \frac{3}{2}.
  2. 2
    Verify: 93/2=(32)3/2=33=279^{3/2} = (3^2)^{3/2} = 3^3 = 27. ✓

Answer

32\frac{3}{2}
Since 9=329 = 3^2 and 27=3327 = 3^3, using base 3 for the change of base makes both logarithms simple integers. The result 32\frac{3}{2} can be verified: 93/2=279^{3/2} = 27.

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 2 medium

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

Example 4 hard

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

← All Change of Base Formula Examples Change of Base Formula Concept