Definition:Exponential/Real/Definition 4

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Consider the natural logarithm $\ln x$, which is defined on the open interval $\left({0 \,.\,.\, +\infty}\right)$.

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 $\R_{>0}$.