Natural Logarithm Function

Definition of the Natural Logarithm Function

• The natural logarithm function, denoted as ln, is defined on the interval $]0; +\infty[$.
• It represents the primitive of the function $x \longmapsto \frac{1}{x}$ that equals zero at 1.

Consequences of the Definition

• The ln function is differentiable on $]0; +\infty[$, and for all $x>0$, its derivative is $\ln'(x) = \frac{1}{x}$.
• This means that the ln function is strictly increasing on $]0; +\infty[$.
• $\ln(1) = 0$.

Algebraic Properties

• For strictly positive real numbers $a$ and $b$, and any integer $n$:
  • $\ln(ab) = \ln(a) + \ln(b)$
  • $\ln(\frac{1}{a}) = -\ln(a)$
  • $\ln(\frac{a}{b}) = \ln(a) - \ln(b)$
  • $\ln(a^n) = n\ln(a)$
  • $\ln(\sqrt{a}) = \frac{1}{2}\ln(a)$

Limits of the Natural Logarithm Function

• $\lim_{x \to +\infty} \ln(x) = +\infty$
• $\lim_{x \to 0} \ln(x) = -\infty$
• $\lim_{x \to 0} x\ln(x) = 0$
• $\lim_{x \to +\infty} \frac{\ln(x)}{x} = 0$

Derivatives Involving the Natural Logarithm

• If $u$ is a differentiable and strictly positive function on an interval I, then the function $x \longmapsto \ln(u(x))$ is differentiable on I.
• The derivative is $(\ln(u(x)))'=\frac{u'(x)}{u(x)}$.
• Example:
  • Given $f(x) = \ln(x^2+1)$,
  • $f'(x) = \frac{2x}{x^2+1}$.