A logarithm answers the question: which power of the base gives this number? The laws of logarithms are the index laws in another form, and each law only works when every logarithm in it is defined.
This lesson belongs to SPM Additional Mathematics indices, surds and logarithms. If indices feel unsteady, begin with applying index laws with fractional powers.
What do the laws say?
The statement log₂ 8 = 3 is the same as 2³ = 8. All the laws below use one base throughout.
| Law | Example |
|---|---|
| log(xy) = log x + log y | log₂ 4 + log₂ 8 = log₂ 32 = 5 |
| log(x ÷ y) = log x − log y | log₃ 18 − log₃ 2 = log₃ 9 = 2 |
| log(xⁿ) = n log x | log₅ 25³ = 3 × 2 = 6 |
| log_a 1 = 0 and log_a a = 1 | log₇ 1 = 0, log₇ 7 = 1 |
| log_a b = log_c b ÷ log_c a | log₄ 32 = 5 ÷ 2 |
Worked example: combining logarithms
Simplify log₂ 12 + log₂ 6 − log₂ 9 to a single number.
Add first, then subtract. Adding logs multiplies the numbers and subtracting divides them.
log₂ 12 + log₂ 6 − log₂ 9 = log₂ (12 × 6 ÷ 9) = log₂ 8 = 3.
The same laws work backwards. If log₂ 3 = p and log₂ 5 = q, then log₂ 45 = log₂ (9 × 5) = 2p + q, and log₂ 0.6 = log₂ (3 ÷ 5) = p − q.
The mistake that costs marks
The slip is using log x² = 2 log x without asking whether x can be negative. The law needs x > 0, since log x has to exist.
Take x = −3. The left side is log₂ 9, which is defined and equals 2 log₂ 3. The right side, 2 log₂ (−3), is undefined.
| Step | Wrong | Right |
|---|---|---|
| Rewrite log₂ x² | 2 log₂ x always | 2 log₂ x only if x > 0 |
| Value x = −3 | Accepted | log₂ x undefined, so the law cannot be used |
| Safe form | (none) | 2 log₂ |x| |
In equations this matters even more. You solve first, then test every answer against the domain: each number inside a logarithm must be positive.
Changing the base
Change of base lets you rewrite a logarithm in a base that suits the question. Find log₄ 32.
Write both 4 and 32 as powers of 2. Then log₄ 32 = log₂ 32 ÷ log₂ 4 = 5 ÷ 2 = 2.5.
Check: 4^2.5 = (2²)^2.5 = 2⁵ = 32.
Check yourself
Given log₃ 2 = m, express log₃ 18 and log₃ (2 ÷ 27) in terms of m.
Answer
18 = 2 × 9, so log₃ 18 = log₃ 2 + log₃ 9 = m + 2.
log₃ (2 ÷ 27) = log₃ 2 − log₃ 27 = m − 3.
Numerical check with m ≈ 0.631: log₃ 18 ≈ 2.631, and 3^2.631 ≈ 18.
What to study next
Put the laws to work on equations in solving exponential and logarithmic equations. Before that, the worksheet on word problem structure helps with setting out multi-step questions.
Test the whole chapter with the chapter practice set. To go through logarithm questions with a teacher, see online one-to-one Additional Mathematics tuition.