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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:

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

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