Key Idea: These are your algebra cheat-sheets for powers and logs — used to simplify expressions and to solve exponential and log equations. Almost all of it is Paper 1 by hand; Paper 2 only adds a graphing shortcut.
⚡ Laws of exponents (Paper 1)
- the common base — must be the same to combine
- negative power = reciprocal
- fractional power = nth root, then mth power
| Operation | Rule (same base) | Example |
|---|---|---|
| Multiply | Add the exponents | x⁵ × x³ = x⁸ |
| Divide | Subtract the exponents | x⁸ ÷ x² = x⁶ |
| Power of a power | Multiply the exponents | (x²)³ = x⁶ |
| Power 0 | Always 1 | a⁰ = 1 |
| Fractional | Root (bottom), then power (top) | 27²/³ = (∛27)² = 9 |
🟰 Laws of logarithms (Paper 1)
- log of 1 is always 0
- log of the base is 1
- change of base — swap to any base you can compute
| Law | Combine (→) | Expand (←) |
|---|---|---|
| Product | log x + log y = log(xy) | Split a product into a sum |
| Quotient | log x − log y = log(x/y) | Split a quotient into a difference |
| Power | m log x = log xᵐ | Bring an exponent down as a coefficient |
| Change of base | logₐ x = log x ÷ log a | Rewrite an awkward base |
| Equation type | Method | Quick example |
|---|---|---|
| aˣ = b, bases match | Equate the exponents | 4ˣ = 8 → 2²ˣ = 2³ → x = 3/2 |
| aˣ = b, bases differ | Take logs (power law drops x) | 5ˣ = 20 → x = log 20 / log 5 |
| logₐ(expr) = c | Convert: expr = aᶜ | ln(x²−16) = 0 → x² − 16 = 1 |
| Two logs = c | Combine to one log, then convert | log₂x + log₂(x−2) = 3 → x = 4 |
✏️ IB-style worked examples
IB-style question — simplify with the index laws
Simplify (p⁵ × p³) ÷ p², leaving your answer as a single power of p.
Step by step:
Multiply — add the exponents.
Divide — subtract the exponent.
p⁶
IB-style question — condense into a single logarithm
Write ln 5 + 2 ln 2 − ln 10 as a single logarithm.
Step by step:
Coefficient first: the power law moves the 2 up.
Add → multiply on top, subtract → divide.
Simplify the inside.
ln 2
IB-style question — solve an exponential by taking logs
Solve 7ˣ = 50, giving the exact value of x.
Step by step:
Bases won't match, so take logs of both sides.
The power law brings x down.
Divide to isolate x.
x = log 50 / log 7 (≈ 2.01)
🔒 GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Important: log(x + y) ≠ log x + log y. The laws act only on a product, quotient or power — never on a sum or difference inside one log. And after solving a log equation, check every argument is positive: discard any root that makes a log's inside ≤ 0.
Tap each card to reveal the answer.
Simplify x⁷ ÷ x³ x⁴ — divide means subtract the exponents, 7 − 3.
Evaluate 16³/⁴ without a calculator 8 — fourth root of 16 is 2, then 2³ = 8.
Write 2 log x + log y as one logarithm log(x²y) — the 2 becomes a power, then add → multiply.
Solve 2ˣ = 16 x = 4 — write 16 = 2⁴, equal bases ⇒ equal exponents.
Solve ln x = 3, exact answer x = e³ — convert to exponential form: x = e³.
Exam tips
- Same base only: multiply → add exponents, divide → subtract, power of a power → multiply.
- Negative power = reciprocal; fractional power m/n = nth root then mth power.
- Logs: product → add, quotient → subtract, power → coefficient (read right-to-left to combine).
- Bases match → equate exponents; otherwise take logs. A lone log = number → convert to expr = aᶜ.
- Reject any solution that makes a log's argument ≤ 0, and use change of base for an awkward base.