Definition: The natural logarithm, written ln x, is the logarithm with base e, where e is approximately 2.71828; it undoes the function e to the x.
What is ln?
The symbol ln x means loge(x), with the irrational constant e approximately equal to 2.71828. It is the inverse of y = ex, so ln(ex) = x and eln x = x for x > 0.
Why is it called natural?
Continuous growth and decay, such as compound interest and radioactive decay, produce the base e on their own. In calculus the derivative of ln x is 1/x, a very simple result, which is why the base feels natural.
How do you use it to solve equations?
Worked example: solve 5e0.4t = 60. Divide by 5 to get e0.4t = 12. Take ln of both sides: 0.4t = ln 12. Since ln 12 is about 2.4849, t = 2.4849 / 0.4, so t is about 6.21.
Special values to remember: ln 1 = 0 and ln e = 1.
What is the usual slip?
Students write ln(x + 3) = ln x + ln 3. The product rule splits ln(3x), never ln(x + 3). Mathbot suggests testing with x = 1 to catch this.
Where to learn it
- Logarithms: The Inverse of Exponents – lesson
- Exponents, Logarithms and Growth Models Explained – lesson