Logarithms and the laws of logs
A logarithm answers the question "what power?". log₂ 8 = 3 because 2³ = 8. The laws of logs turn multiplication into addition and powers into multiplication, which is what lets you solve equations with the unknown in the power.
Part 1 of 3: Learn it
In short
- logₐ x = n means aⁿ = x.
- log x + log y = log(xy); log x − log y = log(x/y); k log x = log(xᵏ).
- To solve aˣ = b, take logs of both sides: x = log b ÷ log a.
Where this is in your specification
Spec points: DfE F3 and F4 (AQA 7357 F3 and F4, Edexcel 9MA0 Pure topic 6, OCR A H240 1.06)
| Board | Topic: Exponentials and logarithms |
|---|---|
| DfE content | F1-F7 |
| AQA 7357 | F1-F7 |
| Edexcel 9MA0 | Pure topic 6 |
| OCR H240 | 1.06 |
| Higher C847 76 | Algebraic and trigonometric skills: exponentials and logarithms |
What a logarithm is
logₐ x is the power you raise a to in order to get x. So log₁₀ 1000 = 3 and log₃ 81 = 4. Two values hold for any base: logₐ a = 1 and logₐ 1 = 0. The natural log, ln x, has base e.
The laws
The laws only work when the logs have the same base. There is no law for log(x + y).
Solving equations
- Unknown in a power: take logs of both sides, bring the power down, and divide.
- Logs on both sides or in a sum: combine into a single log, then rewrite without logs using the definition.
- Always check answers in the original equation: you cannot take the log of zero or a negative number.
Find log₂ 32.
Show the answer
5, because 2⁵ = 32.
Part 2 of 3: See it worked
Worked examples
Example 1
Solve 3ˣ = 20, giving x to 3 significant figures.
- Take logs: x ln 3 = ln 20
- x = ln 20 ÷ ln 3
- = 2.7268...
Answer: x = 2.73.
Example 2
Solve log₂(x + 1) + log₂(x − 1) = 3.
- Combine: log₂[(x + 1)(x − 1)] = 3
- (x + 1)(x − 1) = 2³ = 8, so x² − 1 = 8 and x² = 9
- x = 3 or x = −3; x = −3 would need the log of a negative number
Answer: x = 3.
Example 3
Write 2 log 5 + log 4 as a single number (logs to base 10).
- 2 log 5 = log 25
- log 25 + log 4 = log 100
- log₁₀ 100 = 2
Answer: 2.
Common mistakes
- Writing log(x + y) as log x + log y.
- Dividing logs as log(x/y) when the expression is log x ÷ log y. These are different.
- Keeping a solution that makes the inside of a log negative or zero.
- Combining logs with different bases.
Write log 18 − log 2 as a single logarithm.
Show the answer
log 9.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
Find log₂ 32.
5, because 2⁵ = 32.
Write log 18 − log 2 as a single logarithm.
log 9.
Solve 5²ˣ = 40 to 3 significant figures.
2x = ln 40 ÷ ln 5 = 2.292, so x = 1.15.
Jobs that use this
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
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