User:Keith.U/Exposition of the Natural Logarithm Function/Real

Preamble
The (real) natural logarithm function is a real function and is denoted $\ln$.

Definition

 * $\ln: \R_{>0} \to \R$ can be defined by the following definite integral:


 * $\ln x := \displaystyle \int_{1}^{x} \frac{1}{t} \ \mathrm d t$

Definition

 * $\ln: \R_{>0} \to \R$ can be defined as the inverse mapping of the  natural exponential $\exp$, where $\exp$ is defined as:


 * $\displaystyle \exp x := \sum_{n \mathop = 0}^{\infty} \frac{x^n}{n!} = \lim_{n \to \infty} \left({ 1 + \frac{x}{n} }\right)^{n} = e^{x} = $ "the unique solution to $D \exp = \exp \land \exp 0 = 1$"

Definition

 * $\ln: \R_{>0} \to \R$ can be defined as the limit of the following sequence:


 * $\ln x := \displaystyle \lim_{n \to \infty} n \left({ \sqrt[n]{x} - 1 }\right)$