TeXipedia

ln

Represents the natural logarithm function with base e (Euler's number) in mathematical expressions.

Overview

Essential in mathematical and scientific contexts where exponential relationships need to be analyzed or simplified. Common applications include:

  • Solving exponential equations and growth/decay problems
  • Converting multiplicative relationships to additive ones
  • Analyzing compound interest and continuous growth models
  • Simplifying calculations in calculus, particularly when differentiating or integrating exponential functions
  • Expressing entropy and information theory concepts in physics and computer science

Examples

Natural logarithm of x

y=ln⁡xy = \ln x 
y = \ln x

Solving an exponential equation using natural logarithm

e2x=8  ⟹  2x=ln⁡8e^{2x} = 8 \implies 2x = \ln 8 
e^{2x} = 8 \implies 2x = \ln 8

Natural logarithm in a calculus derivative

ddxln⁡x=1x\frac{d}{dx} \ln x = \frac{1}{x} 
\frac{d}{dx} \ln x = \frac{1}{x}