Overview
Logarithms are the inverse of exponentials. They turn powers into something you can solve.
Definition
If y = aˣ, then x = logₐ y. The natural log ln is logₑ, the inverse of eˣ.
Laws of logarithms
- log a + log b = log(ab)
- log a − log b = log(a ÷ b)
- n log a = log(aⁿ)
- logₐ a = 1, logₐ 1 = 0
Solving equations
Take logs of both sides to bring the power down. For 2ˣ = 20: x log 2 = log 20 → x = log 20 ÷ log 2 ≈ 4.32.
Linearising
A relation like y = k aˣ becomes linear by taking logs: ln y = ln k + x ln a — plot ln y against x to get a straight line (gradient ln a, intercept ln k).
Common mistakes
- Writing log(a + b) = log a + log b — false; only products add.
Exam tips
- ln and eˣ cancel: ln(eˣ) = x and e^(ln x) = x.