Definition and basic properties

Exponential functions and logarithms: Exponential functions

Definition and basic properties

We have already seen that a power function in $x$ consists of a power $x^p$ , for a certain number $p$ . The quadratic function with definition $x^2$ is a simple example. An exponential function also consists of a power, but in this case, the independent variable $x$ is the exponent, such as, for example, in the function definition $2^x$ .

A function of the form $f(x)=a^x$ for $a>0, a\neq 1$ is called an exponential function with base $a$ .

The diagram on the right-hand side shows the graph of $a^x$ for some values of $a$ .

The domain of an exponential function is $\mathbb{R}$ and its range is the open interval $(0,\infty)$ .

Examples

grafieken van exponentiële functies

The number $a=1$ is not useful in the definition of an exponential function because $1^x=1$ for all $x$ and you would preferably call such a function a constant function or a constant linear function. But to make the description of properties of exponential functions easier we will accept 1 as base.

Interactive visualisation By moving the slider in the below interactive figure below you can get an idea how the graph of the exponential function

$f(x)=a^x$ looks like for various values of the base $a$ .

Some properties of an exponential function $f(x)=a^x$ :

$f(0)=1$ (each graph of an exponential function goes through the point (0,1)).
$f(x)>0$ for all $x$ .
$f$ is increasing if and only if $a>1$ . The larger $a$ , the faster the function increases.
$f$ is decreasing if and only if $0<a<1$ . The closer $a$ is to 0, the faster the function decreases.
The horizontal axis is a horizontal asymptote for each exponential function.
If $0<a<1$ , then the function values $a^x$ are small for large positive values of $x$ (in mathematical language: $a^x\to 0$ as $x\to\infty$ , or even more formal, $\lim_{x\to \infty}a^x=0$ ). If $a>1$ , then $\lim_{x\to -\infty}a^x=0$ ).
Exponential functions grow faster than polynomials. In mathematical terminology: voor the base $a>1$ en natural number $n$ there exists a number $N$ such that $a^x>x^n$ for all $x>N$ . You could choose $N=2n+1+\frac{2^n(n+1)!}{(a-1)^{n+1}}$ .

Mathcentre video

Exponential Functions (18:18)