What are the laws of logarithms?
Short answer
The three main laws are: log a + log b = log(ab), log a − log b = log(a ÷ b), and log(aⁿ) = n log a. They hold for any base as long as both logs use the same base. A logarithm answers the question: what power of the base gives this number? So log₂ 8 = 3 because 2³ = 8.
Step by step
- Remember the definition: if y = bˣ then x = log_b y.
- Combine a sum of logs into the log of a product.
- Combine a difference of logs into the log of a quotient.
- Bring powers down in front: log x³ = 3 log x.
- Use log_b b = 1 and log_b 1 = 0 to simplify.
Worked example
Solve log₃ x + log₃ 4 = 2
- log₃ (4x) = 2
- 4x = 3² = 9
- x = 9/4
Common mistakes
- Writing log(a + b) = log a + log b. There is no rule for the log of a sum.
- Combining logs with different bases.
- Forgetting that you can only take the log of a positive number.
Useful for:A LevelWAECBacIBMatric
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