Laws of logarithms and equations
See the product and power laws, solve a logarithmic equation, and straighten data with logarithms.
📖 Start here: read the steps of the “Lesson” in order (1, 2, 3 …) and answer each question. Then continue with “Practice”.
- UK: Year 12
- France: grade 12
- Germany: grade 10
- Iran: grade 11
- Afghanistan: grade 11
- Syria: grade 12
- Palestine: grade 10
- Ukraine: grade 11
- Russia: grade 10
Guide
A logarithm turns multiplying into adding: log₂(8×4) = log₂8 + log₂4. A power comes down as a multiplier: log₂(162) = 2×log₂16. Change of base: log of x to base b equals log x ÷ log b, so a calculator's base 10 will do.
To solve log₂(x + 3) = 4, write it as a power: x + 3 = 24. Careful: do not forget to take 3 away from 16.
If y = a×bx, taking logs gives log y = log a + x×log b. So log y against x is a straight line: its gradient is log b and its height at x = 0 is log a.
Common slips: adding the numbers instead of the logs; adding instead of multiplying by the power; forgetting to move the 3 in the equation; turning the gradient upside down.