Key Idea: A logarithm answers one question: “what power?” It is the inverse of raising to a power, and on Paper 1 you evaluate exact log values by hand.
🔄 The definition: logₐb is a power
- the base (a > 0, a ≠ 1)
- the number you take the log of (b > 0)
- the answer — the power a is raised to
logₐ 1 = 0 (because a⁰ = 1). logₐ a = 1 (because a¹ = a). logₐ (1/b) = −logₐ b — a reciprocal flips the sign.
🔢 log and ln — logs with a hidden base
- base 10 — the “common” log
- base e (e ≈ 2.718) — the “natural” log
✏️ IB-style worked examples
IB-style question — convert between the two forms
(a) Write 2⁶ = 64 in logarithmic form. (b) Write log₄ 64 = 3 in exponent form.
Step by step:
(a) Keep the base 2; the exponent 6 becomes the log value.
(b) Keep the base 4; the log value 3 is the power.
Final answer:
(a) log₂ 64 = 6 (b) 4³ = 64
IB-style question — evaluate a logarithm (Paper 1)
Evaluate log₃ 81 without a calculator.
Step by step:
Ask: 3 to what power gives 81? Write 81 as a power of 3.
The exponent is the answer.
Final answer:
log₃ 81 = 4
IB-style question — a reciprocal and a root (Paper 1)
Evaluate (a) log₂ (1/32) and (b) log₉ 3.
Step by step:
(a) A reciprocal gives a negative power: 1/32 = 2⁻⁵.
(b) A root gives a fractional power: 3 = 9¹/².
Final answer:
(a) −5 (b) ½
Important: logₐ 1 = 0 for every base (because a⁰ = 1) — do not write 1. And logₐ b is not a ÷ b: it is the power that turns a into b. log₂ 8 = 3, not 4.
Tap each card to reveal the answer.
Exam Tips
- aˣ = b ⇔ x = logₐ b — same fact, two forms. The subscript is always the base.
- A log gives the power. To evaluate, write the number as a power of the base.
- Know logₐ 1 = 0 and logₐ a = 1; a reciprocal → negative power, a root → fractional power.
- log x means base 10; ln x means base e.
- Paper 1: evaluate by definition. Paper 2: type log / ln (change of base for other bases, topic 1.7).