Logarithms
A logarithm answers: what power gives this number?
About this topic
A logarithm answers the question "what exponent do I need?" If bˣ = y then log_b(y) = x. So log₂(8) = 3 because 2³ = 8. Logarithms are the inverse of exponential functions, and their graphs are reflections of each other across the line y = x.
The argument must be positive, and log_b(1) = 0 and log_b(b) = 1 for any valid base. Common logs use base 10, so log(1000) = 3. Logarithms compress huge ranges into manageable numbers, which is why the Richter scale for earthquakes, decibels for sound and pH for acidity all use them. An earthquake of magnitude 7 is ten times stronger in amplitude than one of magnitude 6.
Before calculators, logarithm tables helped scientists multiply large numbers by adding, and slide rules were built on this. Today they are used to solve for unknown exponents, such as how long it takes money to double.
Students often think a log is a number multiplied by something, as if log were a variable. Instead read it as a question about powers. The log of a negative number or of zero is undefined in the reals, and log(a + b) is not log a + log b.
Key ideas
- log_b(x) = y means bʸ = x
- log₁₀(1000) = 3
- Logs are inverses of exponentials
Quick check
Use cases: where Logarithms helps
- Homework: review the key idea behind Logarithms before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Logarithms without notes.
- Visual learning: play with the interactive lab to see how Logarithms changes when you adjust its inputs.
- Teaching a friend: explain 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 Logarithms?
A logarithm answers: what power gives this number?
What will I learn in Logarithms?
log_b(x) = y means bʸ = x. log₁₀(1000) = 3. Logs are inverses of exponentials.
Is 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 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 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.