Natural Logarithm
ln x is the logarithm with base e.
About this topic
The natural logarithm ln(x) is the logarithm with base e, so ln(x) = y means eʸ = x. It is the inverse of eˣ, so ln(eˣ) = x and e^(ln x) = x. Also ln(1) = 0 and ln(e) = 1.
It obeys all the usual log properties and is the standard tool for solving equations with e, such as continuous growth and decay. To solve 50e^(0.03t) = 200, divide to get e^(0.03t) = 4, take ln of both sides to get 0.03t = ln 4, so t = ln 4 / 0.03, about 46.2. The time for money to double under continuous growth is ln 2 / r, about 0.693 divided by the rate.
In calculus, the derivative of ln x is 1/x, which gives it a special role. Scientists use natural logs in the formulas for half-life, entropy, learning curves and compound growth, and they appear in the decibel and information theory formulas.
Do not mix up ln and log, which usually means base 10, and remember that ln of a negative number or zero is undefined. The ln button on a calculator is separate from log, so press the right one.
Key ideas
- ln e = 1 and ln 1 = 0
- Inverse of eˣ
- Slope of ln x at x is 1/x
Quick check
Use cases: where Natural Logarithm helps
- Homework: review the key idea behind Natural Logarithm before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Natural Logarithm without notes.
- Visual learning: play with the interactive lab to see how Natural Logarithm changes when you adjust its inputs.
- Teaching a friend: explain Natural Logarithm 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 Natural Logarithm?
ln x is the logarithm with base e.
What will I learn in Natural Logarithm?
ln e = 1 and ln 1 = 0. Inverse of eˣ. Slope of ln x at x is 1/x.
Is Natural Logarithm 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 Natural Logarithm 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 Natural Logarithm?
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.