Definition:Exponential Function/Real/Inverse of Natural Logarithm
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Definition
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 number $\exp x$ is called the exponential of $x$.
The domain of $\exp$ is $\R$, and the codomain of $\exp$ is $\R_{>0}$.
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 3.3$. Injective, surjective, bijective; inverse mappings: Example $48$
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 14.4$
- Weisstein, Eric W. "Exponential Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ExponentialFunction.html