Properties
A standard method from Exponential and logarithmic functions: the steps and worked examples.
Steps
| Rule | Why |
|---|---|
| logₐ1 = 0, logₐa = 1 | a⁰ = 1, a¹ = a |
| log MN = log M + log N | multiplying powers adds the indices |
| log (M/N) = log M − log N | dividing subtracts |
| log Mᵏ = k log M | (aˣ)ᵏ = a^(kx) |
| log_b N = logₐN / logₐb | change of base, so the calculator's log (base 10) can be used |
Not rules: log(M + N) ≠ log M + log N; (log M)/(log N) ≠ log(M/N).
- Write each term as the log of a single number or letter, with any multiplier in front.
- Apply one rule from the table per line: move multipliers up as powers, then combine sums and differences.
- Check you have not used one of the "not" rules above.
Worked examples
Example 1: Use the properties of logarithms: write log 12 − log 3 + 2 log 5 as a single log, then evaluate it.
2 log 5 = log 5² = log 25
log 12 − log 3 + log 25 = log (12 ÷ 3 × 25)
= log 100
= 2For the whole topic, read Exponential and logarithmic functions: key points, common mistakes and how the DSE tests it.