What Is Natural Logarithm?
The natural logarithm ln(x) is the logarithm with base e, where e ≈ 2.71828 is Euler's number. It answers the question "to what power must e be raised to obtain x?" Because e arises naturally in continuous growth and change, ln is the "natural" logarithm used throughout calculus, physics, and finance.
ln is only defined for positive inputs, and it increases slowly for large numbers: ln(1) = 0, ln(e) = 1, and ln(x) turns multiplication into addition. It is the inverse of the exponential function, so e^(ln x) = x for every positive x.
Formula
ln(x) = log_e(x), with e ≈ 2.71828
ln(1) = 0, ln(e) = 1
ln(a · b) = ln(a) + ln(b)
ln(aᵏ) = k · ln(a)
Applications
- Solving for time in continuous growth and decay problems
- Integrating functions of the form 1/x
- Linearizing exponential data for analysis and plotting
- Computing continuously compounded interest and returns
- Modeling cooling, charging, and relaxation processes
Sources
- Stewart, J. (2015). Calculus: Early Transcendentals (8th ed.). Cengage Learning.
- Abramowitz, M., & Stegun, I. A. (1964). Handbook of Mathematical Functions. National Bureau of Standards.