Change of Base

This lesson covers the change of base formula for logarithms, which allows you to convert between different logarithm bases. This is needed for the Edexcel 9MA0 specification and is useful for solving equations, evaluating logarithms on a calculator, and proving results.

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The Change of Base Formula

The change of base formula states:

log_a(x) = log_b(x) / log_b(a)

This allows you to express a logarithm in any base b. The most common choices are b = 10 (common logarithm) and b = e (natural logarithm).

Proof

Let log_a(x) = n. Then aⁿ = x.

Take log_b of both sides: log_b(aⁿ) = log_b(x)

By the power law: n × log_b(a) = log_b(x)

Therefore: n = log_b(x) / log_b(a)

Since n = log_a(x), we have: log_a(x) = log_b(x) / log_b(a)

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Using the Formula with a Calculator

Calculators typically have buttons for log₁₀ and ln, but not for other bases. The change of base formula lets you evaluate any logarithm.

Example 1: Find log₃(20)

• log₃(20) = log₁₀(20) / log₁₀(3) = 1.3010 / 0.4771 ≈ 2.727

Or using natural logarithms:

• log₃(20) = ln(20) / ln(3) = 2.9957 / 1.0986 ≈ 2.727