Natural Logarithm (ln) Function

• The natural logarithm function, denoted as ln, is defined on $]0; +\infty[$.
• ln is the primitive of the function $x \mapsto \frac{1}{x}$ that vanishes at 1.

Properties of ln

• $\ln(1) = 0$
• $\ln(e) = 1$
• For all $x > 0$, $\ln'(x) = \frac{1}{x}$

Variations of ln

• The ln function is strictly increasing on $]0; +\infty[$.

Algebraic Properties

• For all real numbers $a > 0$ and $b > 0$, and for any integer n:
  • $\ln(ab) = \ln(a) + \ln(b)$
  • $\ln(\frac{a}{b}) = \ln(a) - \ln(b)$
  • $\ln(\frac{1}{b}) = -\ln(b)$
  • $\ln(a^n) = n\ln(a)$
  • $\ln(\sqrt{a}) = \frac{1}{2}\ln(a)$

Common/Usual Limits

• $\lim_{x \to +\infty} \ln(x) = +\infty$
• $\lim_{x \to 0} \ln(x) = -\infty$
• $\lim_{x \to +\infty} \frac{\ln(x)}{x} = 0$
• $\lim_{x \to 0} x\ln(x) = 0$
• $\lim_{x \to 1} \frac{\ln(x)}{x-1} = 1$
• $\lim_{h \to 0} \frac{\ln(1+h)}{h} = 1$

Derivatives

• If u is a differentiable and strictly positive function on an interval I, then $f = \ln(u)$ is differentiable on I and $f' = \frac{u'}{u}$.

Examples of derivatives:

• $f(x) = \ln(2x+1)$ then $f'(x) = \frac{2}{2x+1}$
• $f(x) = \ln(x^2+1)$ then $f'(x) = \frac{2x}{x^2+1}$
• $f(x) = \ln(\sqrt{x})$ then $f'(x) = \frac{1}{2x}$

Equations

• For all real numbers a and b strictly positive: $\ln(a) = \ln(b) \Leftrightarrow a = b$

Inequalities

• For all real numbers a and b strictly positive:
  • $\ln(a) < \ln(b) \Leftrightarrow a < b$
  • $\ln(a) > \ln(b) \Leftrightarrow a > b$