Key Idea: Exponents and logarithms are inverses — one undoes the other. The index laws let you simplify expressions with powers. Logarithms solve equations where the unknown is in the exponent. These tools appear in exponential growth models, solving for time, and simplifying complex expressions.
Three things IB tests on this topic:
📐 Index laws
🔄 The log ↔ exponential connection
Method: Take log of both sides, then use the power rule to bring the exponent down. 3ˣ = 50 log(3ˣ) = log 50 x log 3 = log 50 x = log 50 ÷ log 3 = 3.56 (3 s.f.) The power rule turns the unknown exponent into a multiplier — that is why it works.
✏️ Worked examples
Simplify with index laws
Simplify: (2x³)² ÷ x
Step by step:
Power of a product: (2x³)² = 4x⁶
Divide: 4x⁶ ÷ x = 4x⁶⁻¹
Answer: 4x⁵
Final answer:
4x⁵
Solve an exponential equation
Solve: 5ˣ = 80
Step by step:
Take log of both sides: log(5ˣ) = log 80
Power rule: x log 5 = log 80
Divide: x = log 80 ÷ log 5
Calculate: x = 1.903 ÷ 0.699 = 2.72 (3 s.f.)
Final answer:
x ≈ 2.72
Use log laws to simplify
Write log 6 + log 5 − log 3 as a single value.
Step by step:
Product rule: log 6 + log 5 = log(6 × 5) = log 30
Quotient rule: log 30 − log 3 = log(30 ÷ 3) = log 10
log 10 = 1
Final answer:
1
Index laws only apply when the base is the same. 2³ × 3⁴ cannot be simplified — different bases. Log laws only apply when the base is the same. Never mix log (base 10) and ln (base e) in the same calculation. Quick values to know: log 1 = 0, log 10 = 1, ln 1 = 0, ln e = 1. Paper 2: After solving an exponential equation, verify by substituting back: 5^2.72 ≈ 80 ✓