User:Keith.U/Exposition of the Natural Logarithm Function/Real
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Preamble
The (real) natural logarithm function is a real function and is denoted $\ln$.
Approach 1: Integral Definition
Definition
- $\ln: \R_{>0} \to \R$ can be defined by the following definite integral:
- $\ln x := \ds \int_1^x \frac 1 t \rd t$
Approach 2: Inverse of $\exp$
Definition
- $\ln: \R_{>0} \to \R$ can be defined as the inverse mapping of the natural exponential $\exp$, where $\exp$ is defined as:
- $\ds \exp x := \sum_{n \mathop = 0}^\infty \frac {x^n} {n!} = \lim_{n \mathop \to \infty} \paren {1 + \frac x n}^n = e^x = $ "the unique solution to $D \exp = \exp \land \exp 0 = 1$"
Approach 3: Limit of Sequence
Definition
- $\ln: \R_{>0} \to \R$ can be defined as the limit of the following sequence:
- $\ln x := \ds \lim_{n \mathop \to \infty} n \paren {\sqrt [n] x - 1}$