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Natural Logarithm

Logarithm at base "e" is known as natural logarithm, where "e" is an irrational number whose value is 2.718281828.... Natural logarithms are represented by $log_{e}\ x$ or more commonly ln x. We may define natural logarithm as illustrated below:

Natural Logarithm
We have following two important identities related to natural logs:
  • $\ln e^{x}=x$
  • $e^{\ln x}=x$ (For every $x \ > 0$)
Natural logarithm also follows laws of common logarithm (for $ u,\ v\ > 0$), as:
  • $ln (uv)=ln\ u+ln\ v$
  • $ln ($\frac{u}{v}$)=ln\ u-ln\ v$
  • $ln\ u^{v}=v ln\ u$
Other important properties of natural logarithm are listed below:
  • Natural log of 1: ln 1 = 0.
  • Natural log of -1: $ln(-1)=i\pi $
  • Limits on natural log: $\lim_{x\rightarrow 0}\ln x=-\infty $ and $\lim_{x\rightarrow \infty }\ln x=\infty $.
  • Derivative: $\frac{\mathrm{d}}{\mathrm{d} x}$ $\ln x=$$\frac{1}{x}$
  • Integral: $\int \ln x=x\ln x-x+c$, where c is a constant.
  • Taylor series: $\ln(x)= (x - 1) - $$\frac{(x-1) ^ 2}{2}$ + $\frac{(x-1)^3}{3}$ - $\frac{(x-1)^4}{4}$$ +.... $, where 0 < x $\leq$ 2.
Graph of Natural Logarithm
Natural Logarithm Graph

Related Calculators
Natural Logarithm Calculator Calculate Logarithm
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More topics in  Natural Logarithm
Properties of Natural Logarithms
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