The Natural Logarithm
Simple Explanation
The natural logarithm is a logarithm with base e, an irrational constant approximately equal to 2.71828. It is written ln(x), and ln(x) = log_e(x) β the two notations mean the same thing.
Why Do We Need It?
The constant e arises naturally in any process that grows or decays continuously (rather than in discrete steps) β the natural logarithm is the essential tool for working with such processes in calculus and science.
Formula
Natural Logarithm and Euler's Number
ln x = log_e x, where e β 2.71828...
The natural logarithm is simply a logarithm with base e, a special irrational constant that arises naturally in continuous growth and decay processes.
- x
- β the argument of the natural logarithm (x > 0)
- e
- β Euler's number, an irrational constant approximately equal to 2.71828
When to use it: Whenever a process involves continuous (rather than step-by-step) growth or decay, such as continuous compound interest or radioactive decay.
Worked Example
Evaluate a natural logarithm of a power of e
Evaluate ln(eβ΅).
Why Does This Work?
By the definition of a logarithm, log_e(eβΏ) always equals n directly, since eβΏ is already written as e raised to the exact exponent the logarithm is asking for.
Real-Life Example
Continuous compound interest
A savings formula for continuous compounding is A = AβeΚ³α΅.
To solve this formula for the time t, you take the natural logarithm of both sides β ln is the natural partner to any formula built on the constant e.
Practice
Evaluate ln(eΒ³).
MediumCommon mistake
Confusing "ln" with "log" (base 10) β they are logarithms with different bases (e versus 10) and give different numeric results for the same argument, except at x = 1, where both always equal 0.
Quick Review
- ln(x) = log_e(x), the logarithm with base e β 2.71828.
- ln(eβΏ) = n, directly by the definition of a logarithm.
- The natural logarithm is essential for continuous growth/decay processes.