What is an exponential function?

Mar 23, 2015

The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change in the dependent variable. The function is often written as exp(x) It is widely used in physics, chemistry, engineering, mathematical biology, economics and mathematics.

Mar 23, 2015

An exponential function is a function of the form $f \left(x\right) = {a}^{x}$ with $a > 0$ but $a \ne 1$.

For integer and rational $x$, we give definitions of ${a}^{x}$ earlier in algebra classes.
For irrational $x$ we owe you a definition, but one approach to the definition is to describe ${a}^{x}$ for irrational $x$ as the number thar ${a}^{r}$ gets closer to as rational $r$ get close to $x$. (We owe you a proof that there is a unique such number.)

Examples:

$f \left(x\right) = {2}^{x}$

$f \left(x\right) = {5}^{x}$

$f \left(x\right) = {\left(\frac{2}{5}\right)}^{x}$

$f \left(x\right) = {4}^{x} = {\left({2}^{2}\right)}^{x} = {2}^{2 x}$

The last example illustrates why we also consider $f \left(x\right) = {a}^{k x}$ for constant $k \ne 0$ to be exponential functions

We can write $f \left(x\right) = {a}^{k x} = {\left({a}^{k}\right)}^{x}$ and .for $a > 0$ and $k \ne 0$ this $f \left(x\right) = {b}^{x}$ for the "right kind" of $b$. ($b > 0$, and $b \ne 1$)