### 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**

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 \(a\).

Properties 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)).
- \(f(x)>0\) 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)