Properties of Logarithms
Logs turn multiplication into addition.
About this topic
Logarithm properties turn multiplication into addition and powers into multiplication. The product rule log(xy) = log x + log y, the quotient rule log(x/y) = log x − log y, and the power rule log(xᵏ) = k log x all follow from the exponent rules.
The change-of-base formula log_b(x) = log(x)/log(b) lets you compute any log on a calculator. As an example, log₂(5) = log 5 / log 2, about 2.32. You can condense or expand expressions: log(8x) = log 8 + log x, and 2 log 3 + log 4 = log(9 · 4) = log 36. Since the rules mirror exponent laws, adding the exponents when multiplying becomes adding logs.
These rules are used to solve exponential equations, to simplify formulas in chemistry and physics, and to transform curved data into straight lines on log scales, a favorite trick in science.
The classic errors are saying log(x + y) = log x + log y and log(x)/log(y) = log(x/y). Neither is true. Use the rules only for products, quotients and powers inside a single log, and keep the base the same throughout.
Key ideas
- log(ab) = log a + log b
- log(a/b) = log a − log b
- log(aⁿ) = n log a
Quick check
Use cases: where Properties of Logarithms helps
- Homework: review the key idea behind Properties of Logarithms before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Properties of Logarithms without notes.
- Visual learning: play with the interactive lab to see how Properties of Logarithms changes when you adjust its inputs.
- Teaching a friend: explain Properties of Logarithms in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Properties of Logarithms?
Logs turn multiplication into addition.
What will I learn in Properties of Logarithms?
log(ab) = log a + log b. log(a/b) = log a − log b. log(aⁿ) = n log a.
Is Properties of Logarithms part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Properties of Logarithms lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Properties of Logarithms?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.