# 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 $f \paren x = \exp x$ 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 $\left({0 \,.\,.\, +\infty}\right)$.

### Exponential of Natural Logarithm

$\forall x > 0: \exp \left({\ln x}\right) = x$
$\forall x \in \R: \ln \left({\exp x}\right) = x$

### Exponential Function is Continuous

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