#### 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 $a>0$:

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. ${D}_{f}=\mathbb{R}$, whereas the range only consists of the positive real numbers, i.e. ${W}_{f}=(0;\infty )$, 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 $g:\mathbb{R}\to (0;\infty )$, $x\u27fcg(x)={a}^{x}$ for different values of $a$:

- All these exponential functions pass trough the point $(x=0,y=1)$, since $g(x=0)={a}^{0}$ 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.

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 $f(t)=500\xb7{2}^{(t/13)}$ that occurs in Example 6.4.1.