#### Chapter 6 Elementary Functions

Section 6.4 Exponential and Logarithmic Functions

# 6.4.2 Contents

In the previous example, an exponential function with base $a=2$ occurs, and the independent variable - in this example this is the variable $t$ - occurs in the exponent. We will now specify the general mapping rule for an exponential function with an arbitrary base $a$; however, we here assume $a>0$:

$f:\mathrm{ }\left\{\begin{array}{ccc}\hfill ℝ& \hfill \to \hfill & \left(0;\infty \right)\hfill \\ \hfill x& \hfill ⟼\hfill & f\left(x\right)={f}_{0}·{a}^{\lambda x}\hfill \end{array}$

Here, ${f}_{0}$ and $\lambda$ denote so-called parameters of the exponential function that will be introduced below.
The domain of all exponential functions is the set of all real numbers, i.e. ${D}_{f}=ℝ$, whereas the range only consists of the positive real numbers, i.e. ${W}_{f}=\left(0;\infty \right)$, since every power of a positive number can only be positive.
##### Exercise 6.4.2
Why it is assumed that the base $a$ of the exponential function is greater than zero?

Some general properties can be seen from the figure below showing exponential functions $g:ℝ\to \left(0;\infty \right)$, $x⟼g\left(x\right)={a}^{x}$ for different values of $a$:
• All these exponential functions pass trough the point $\left(x=0,y=1\right)$, since $g\left(x=0\right)={a}^{0}$ and ${a}^{0}=1$ for every number $a$.
• If $a>1$, then the graph of $g$ rises from left to right (i.e. for increasing $x$-values); one also says that the function $g$ is strictly increasing. The greater the value of $a$, the steeper the graph of $g$ rises for positive values of $x$. Moving towards ever larger negative values of $x$ (i.e. approaching from right to left) the negative $x$-axis is an asymptote of the graph.
• If $a<1$, then the graph of $g$ falls from left to right (i.e. for increasing $x$-values); one also says that the function $d$ is strictly decreasing. The greater the value of $a$, the slower the graph of $g$ falls for negative values of $x$. Moving towards ever larger positive values of $x$ (i.e. approaching from left to right) the positive $x$-axis is an asymptote of the graph.
What are the parameters ${f}_{0}$ and $\lambda$? The parameter ${f}_{0}$ is easily explained: if the value $x=0$ is inserted in the general exponential function $f$, resulting in

$f\left(x=0\right)={f}_{0}·{a}^{\lambda ·0}={f}_{0}·{a}^{0}={f}_{0}·1={f}_{0} ,$

then it can be seen that ${f}_{0}$ is a kind of starting point or initial value (at least if the variable $x$ is taken for a time); the exponential progression ${a}^{\lambda x}$ is generally multiplied by the factor ${f}_{0}$ and thus weighted accordingly, i.e. stretched (for $|{f}_{0}|>1$) or compressed (for $|{f}_{0}|<1$).
The parameter $\lambda$ that occurs in the exponent is called growth rate; it determines how strong the exponential function - with the same base - increases (for $\lambda >0$) or decreases (for $\lambda <0$). The expression ${a}^{\lambda x}$ is called growth factor.
##### Exercise 6.4.3
Explain the form of the exponential function $f\left(t\right)=500·{2}^{\left(t/13\right)}$ that occurs in Example 6.4.1.