Laws of logarithms

International GCSE · 4PM1 · 1B
About 15 minutes

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Three laws let you combine, split and simplify logarithms. They come straight from the laws of indices.

The three laws

log_a (xy) = log_a x + log_a y
log_a (xy) = log_a x − log_a y
log_a (x^k) = k log_a x

Two special values

log_a 1 = 0, because a⁰ = 1
log_a a = 1, because a¹ = a

Combining into one log

Deal with any number in front first, using the power law. Then add or subtract.
2 log₃ 6 − log₃ 4 = log₃ 36 − log₃ 4 = log₃ 9 = 2

Worked example

Write 2 log₁₀ 5 + log₁₀ 4 as a single logarithm and evaluate it.

2 log₁₀ 5 = log₁₀ 25
log₁₀ 25 + log₁₀ 4 = log₁₀ 100
10² = 100

Answer: 2

Key idea

Adding logs multiplies the numbers, subtracting divides them, and a number in front becomes a power. log_a 1 = 0 and log_a a = 1.

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