Properties of Exponential Function

Theorem

Let $x \in \R$ be a real number.

Let $\exp x$ be the exponential of $x$.

Then:

Exponential of Zero

$\exp 0 = 1$

Exponential of One

$\exp 1 = e$

where $e$ is Euler's number: $e = 2.718281828\ldots$

Exponential is Strictly Increasing

The function $\map f x = \exp x$ is strictly increasing.

Exponential is Strictly Convex

The function $f \left({x}\right) = \exp x$ is strictly convex.

Exponential Tends to Zero and Infinity

$\exp x \to +\infty$ as $x \to +\infty$

$\exp x \to 0$ as $x \to -\infty$

Thus the exponential function has domain $\R$ and image $\openint 0 \infty$.

Exponential of Natural Logarithm

$\forall x > 0: \map \exp {\ln x} = x$

$\forall x \in \R: \map \ln {\exp x} = x$

Exponential Function is Continuous

$\forall c \in \R: \ds \lim_{x \mathop \to c} \exp x = \exp c$