Change of Base Formula Math Example 4

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

Example 4

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

Solution

  1. 1
    By change of base: log⁡a(b)=ln⁡bln⁡a\log_a(b) = \frac{\ln b}{\ln a} and log⁡b(c)=ln⁡cln⁡b\log_b(c) = \frac{\ln c}{\ln b}.
  2. 2
    Multiply: ln⁡bln⁡a⋅ln⁡cln⁡b=ln⁡cln⁡a=log⁡a(c)\frac{\ln b}{\ln a} \cdot \frac{\ln c}{\ln b} = \frac{\ln c}{\ln a} = \log_a(c).

Answer

log⁡a(b)⋅log⁡b(c)=log⁡a(c)(proven)\log_a(b) \cdot \log_b(c) = \log_a(c) \quad \text{(proven)}
This chain rule for logarithms shows that logarithms of different bases are connected. The proof relies on the change of base formula, which converts all logarithms to natural logarithms, allowing the intermediate base bb to cancel.

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 3 medium

Simplify [formula] to an exact fraction.

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