User:Keith.U/Whatever
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Consider the natural logarithm $\ln x$, which is defined on the open interval $\openint 0 {+\infty}$.
From Logarithm is Strictly Increasing:
- $\ln x$ is strictly increasing.
From Inverse of Strictly Monotone Function:
- the inverse of $\ln x$ always exists.
The inverse of the natural logarithm function is called the exponential function, which is denoted as $\exp$.
Thus for $x \in \R$, we have:
- $y = \exp x \iff x = \ln y$
The exponential function can be defined as a power series:
- $\exp x := \ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}$