The big idea: A logarithm answers the question "what power?".
loga b is the power you raise a to, to get b.
For example, log₂ 8 = 3, because 2³ = 8.
- the base (a > 0, a ≠ 1)
- the result (b > 0)
- the power — i.e. the logarithm
Logs undo powers: ax = b and x = loga b are the same fact written two ways.
Taking a log undoes raising to a power, and vice versa — they are inverses.
Free preview
This is the free notes preview
You're reading the free notes. Aimnova Pro unlocks the full study experience — and you can try it with your first topic free to keep:
- FlashcardsLock in vocabulary and key terms with spaced repetition.
- Practice questionsAnswer exam-style questions and get instant AI marking.
- Mock exams & past-paper vaultSit full mocks and see exactly how examiners award marks.
- Personalised study planA daily plan built around your exam date and weak areas.
Start Studying Free Full access to Aimnova Pro · cancel anytime
Same fact, two forms: To convert, keep the base as the base; the log equals the exponent:
ax = b ⇔ loga b = x.
IB-style question — exponent → log
Write 3⁴ = 81 in logarithmic form.
Step by step
- The base 3 stays the base; the exponent 4 becomes the value of the log.
Final answer
log₃ 81 = 4.
IB-style question — log → exponent
Write log₅ 125 = 3 in exponent form.
Step by step
- The base 5 stays the base; the log value 3 is the exponent.
Final answer
5³ = 125.
Spot the base: The small subscript is the base — and it is the base in both forms.
The log value is always the power.
See how examiners mark answers
Answer exam-style questions with model answers. Learn exactly what earns marks and what doesn't.
Try Exam Vault FreeYour first topic is free to keep • No credit card required
Two logs written without a base: log x means log₁₀ x (base 10).
ln x means loge x (base e, where e ≈ 2.718).
They are just logs with a hidden base.
Quick values: log 1000 = 3 (because 10³ = 1000).
ln e = 1 (because e¹ = e), and ln 1 = 0.