Key Idea: Exponents raise a number to a power. Logarithms undo that — they tell you what the power was. The two are inverses of each other.
Key skills for this topic
Switch log ↔ power
- aˣ = b ↔ x = logₐ b
- log₃ 81 = 4 because 3⁴ = 81
- log = base 10, ln = base e
Common mistakes
- Mixing log and ln in one calc
- Forgetting log laws need same base
- Treating log(A + B) as log A + log B
Solve aˣ = b
- Use: x = log b ÷ log a
- E.g. 3ˣ = 50
- x = log 50 ÷ log 3 ≈ 3.56
| Index law | Rule | Example |
|---|---|---|
| Multiply (same base) | aˣ × aʸ = aˣ⁺ʸ | 2³ × 2⁴ = 2⁷ = 128 |
| Divide (same base) | aˣ ÷ aʸ = aˣ⁻ʸ | 3⁵ ÷ 3² = 3³ = 27 |
| Power of a power | (aˣ)ʸ = aˣʸ | (2³)² = 2⁶ = 64 |
| Zero exponent | a⁰ = 1 | 7⁰ = 1 |
| Negative exponent | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
| Fractional exponent | a¹/ⁿ = ⁿ√a | 8¹/³ = ∛8 = 2 |
🔄 Log ↔ exponential
Key Idea: log₂ 8 = 3 because 2³ = 8. log (no base) = base 10. ln = base e ≈ 2.718.
| Log law (same base) | Rule | Example |
|---|---|---|
| Product | log(A × B) = log A + log B | log 6 + log 5 = log 30 |
| Quotient | log(A ÷ B) = log A − log B | log 30 − log 3 = log 10 = 1 |
| Power | log(Aⁿ) = n log A | log 5³ = 3 log 5 |
🔑 Solving aˣ = b
Example: solve 3ˣ = 50 1. Take log of both sides: log(3ˣ) = log 50 2. Power rule drops x to the front: x · log 3 = log 50 3. Divide both sides by log 3: x = log 50 ÷ log 3 4. On the GDC: x ≈ 3.56 (3 s.f.)
✏️ 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
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.)
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
1
💡 Test yourself — tap to reveal
Evaluate log₂ 32 Ask: 2 to what power = 32? 2⁵ = 32 Answer: 5
Solve 2ˣ = 1000 x = log 1000 ÷ log 2 = 3 ÷ 0.301 Answer: x ≈ 9.97 (3 s.f.)
Simplify log 200 − log 2 Quotient rule: log(200 ÷ 2) = log 100 Answer: 2
What is log₅ 1? Any non-zero base to the power 0 = 1. Answer: 0
Rewrite 4³ = 64 in log form Base 4, result 64, power 3. Answer: log₄ 64 = 3
🎯 IB-style practice — logarithms in context
Key Idea:
- LOG → evaluates log₁₀ (use when finding the dB / pH value).
- 2nd + LOG → 10ˣ (use to undo a log and find the original value).
Part (i) — find loudness from intensity
The loudness of a sound (in dB) is given by <strong>L = 10 log₁₀(I / I₀)</strong>, where I is the intensity in W m⁻² and I₀ = 10⁻¹² W m⁻² is the reference intensity.<br><br>A quiet library reading room has sound of intensity I = 5 × 10⁻⁹ W m⁻². Find its loudness.
Step by step:
Plug in I = 5 × 10⁻⁹ and I₀ = 10⁻¹². Drop the values into the model:
Divide powers of 10 by subtracting exponents: −9 − (−12) = 3:
Type into the GDC with LOG, then round to 3 s.f.:
L ≈ 37.0 dB
🔒 GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Part (ii) — find intensity from loudness
Using the same model <strong>L = 10 log₁₀(I / I₀)</strong> with I₀ = 10⁻¹² W m⁻²:<br><br>A motorcycle engine produces sound of loudness L = 88 dB. Find its intensity I. Give your answer in the form a × 10ᵏ where 1 ≤ a < 10 and k is an integer.
Step by step:
Set L = 88. Put the given loudness into the model — now I is the unknown:
Get rid of the 10 in front of the log. Divide both sides by 10:
Undo the log with 10ˣ. Because log₁₀ and 10ˣ are inverses, the log peels away:
Isolate I by multiplying both sides by 10⁻¹², then combine powers (10ᵃ × 10ᵇ = 10ᵃ⁺ᵇ):
Write in standard form and round to 3 s.f.:
I ≈ 6.31 × 10⁻⁴ W m⁻²
🔒 GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Same base only. 2³ × 3⁴ can't be combined — different bases. Same rule for log laws. Don't mix log and ln in one calculation. Memorise: log 1 = 0, log 10 = 1, ln 1 = 0, ln e = 1. Paper 2 check: after solving aˣ = b, plug your answer back in to verify (e.g. 5².⁷² ≈ 80 ✓).
IB-style question — solving an exponential equation [5 marks]
An investment of $2000 grows by 6% each year, so after t years its value is V = 2000 × 1.06ᵗ dollars. (a) Find the value of the investment after 10 years. (b) Find the number of complete years it takes for the investment to first exceed $5000.
Step by step:
(a) Substitute t = 10 into the model.
(b) Set V > 5000 and divide both sides by 2000 to isolate the power.
The unknown is in the exponent, so take logs of both sides; the power law brings t down.
Evaluate the quotient on the GDC.
t must be a whole number of years and the total must exceed 5000, so round UP to the next year.
(a) $3581.70 (to the nearest cent), about $3580. (b) 16 years.
🔒 GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.