Logarithm Rules: Product, Quotient, and Power
The product, quotient, and power rules for logarithms with two worked examples.
The rules
- log(ab) = log a + log b
- log(a/b) = log a − log b
- log(aⁿ) = n · log a
Logs turn multiplication into addition and exponents into multiplication.
Example 1
Write log₂ 8 + log₂ 4 as a single log, then evaluate.
- log₂(8 × 4) = log₂ 32
- 2⁵ = 32
5
Example 2
Simplify log₃ 81 − log₃ 9.
- log₃(81 ÷ 9) = log₃ 9
- 3² = 9
2
Common mistake
log(a + b) = log a + log b
There is no rule for the log of a sum.
The rules need the same base on every log.
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