Maths
Logarithms & exponentials — quick study summary
GCSE MathsA-Level MathsAP PrecalculusIB Maths
A logarithm answers 'what power do I raise the base to, to get this number?'. log_b(x) = y means bʸ = x. The natural log ln is log base e (e ≈ 2.718). Log laws: log(xy) = log x + log y; log(x/y) = log x − log y; log(xⁿ) = n log x. Logs convert multiplication into addition — that's how slide rules worked, and why log scales handle huge ranges (pH, decibels, earthquake magnitudes). Exponentials and logs are inverses: e^(ln x) = x.
Key points
- log_b(x) = y ⟺ bʸ = x
- log(xy) = log x + log y; log(x/y) = log x − log y; log(xⁿ) = n log x
- log_b(b) = 1; log_b(1) = 0
- Natural log: ln = log_e where e ≈ 2.718
- ln(eˣ) = x and e^(ln x) = x (inverse functions)
Practice quiz
Click each question to reveal the answer.
1. What is log₁₀(1000)?
- 1
- 2
- 3
- 10
Answer: 3
10³ = 1000, so log₁₀(1000) = 3.
2. Simplify: log(2) + log(50)
Answer: log(100) = 2
log x + log y = log(xy), so log(2) + log(50) = log(100) = 2 (assuming log base 10).
3. Solve for x: 2ˣ = 16
Answer: x = 4
16 = 2⁴, so x = 4. Or take log: x = log₂(16) = 4.
Last reviewed: May 2026