The Laws of Logarithms
Turn products into sums, quotients into differences and powers into multipliers, and combine several logarithms into one before solving or simplifying.
What a learner can do afterwards
- Write log x + 2 log y as a single logarithm
- Expand log(a²b / c) into separate terms
- Use a change of base to evaluate a logarithm the calculator has no key for
1 · Read
You already know a log asks which power gives a number. The three laws all follow from that idea. Adding two logs means multiplying what sits inside them. Subtracting means dividing what sits inside. A multiplier in front dives inside as a power.
Take log x + 2 log y. Pull the 2 inside first, so 2 log y becomes log(y^2). Then the plus fuses the pair into log(x y^2). The same two moves turn 2 log x - log y into log(x^2/y).
Take log(a^2 b/c). Split the division first to get log(a^2 b) - log c. Split the product next to get log(a^2) + log b - log c. Bring the power down last to get 2 log a + log b - log c. Read backwards, the same laws condense: 2 log a + log b - log c fuses into log(a^2 b/c).
Some bases have no calculator key, so rewrite with base 10 logs. Log base 2 of 8 becomes log(8)/log(2), and since 2^3 = 8 the value is 3. Log base 4 of 8 becomes log(8)/log(4), which is 1.5, since 4^1.5 = 8. Pick one base and stay with it for the whole question.
Pull powers inside first, then turn plus into multiply and minus into divide, and change base with a division of logs.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.