2 Exponential functions

For a fixed value of the base $a$ the expression $a^x$ clearly varies with the value of $x$ : it is a function of $x$ . We show in Figure 1 the graphs of ${\left(0.5\right)}^x,{\left(0.3\right)}^x,1^x,2^x$ and $3^x$ .

The functions $a^x$ (as different values are chosen for $a$ ) are called exponential functions . From the graphs we see (and these are true for all exponential functions):

If $a>b>0$ then

$a^x>b^x\text{if}x>0\text{and}{a}^x<b^x\text{if}x<0$

Figure 1 :

The most important and widely used exponential function has the particular base $\text{e}=2.7182818\dots$ . It will not be clear to the reader why this particular value is so important. However, its importance will become clear as your knowledge of mathematics increases. The number $e$ is as important as the number $\pi$ and, like $\pi$ , is also irrational. The approximate value of $e$ is stored in most calculators. There are numerous ways of calculating the value of $\text{e}$ . For example, it can be shown that the value of $\text{e}$ is the end-point of the sequence of numbers:

${\left(\frac{2}{1}\right)}^1,{\left(\frac{3}{2}\right)}^2,{\left(\frac{4}{3}\right)}^3,\dots ,{\left(\frac{17}{16}\right)}^{16},\dots ,{\left(\frac{65}{64}\right)}^{64},\dots$

which, in decimal form (each to 6 d.p.) are

$2.000000,2.250000,2.370370,\dots ,2.637929,\dots ,2.697345,\dots$

This is a slowly converging sequence. However, it does lead to a precise definition for the value of $\text{e}$ :

$\text{e}=\underset{n\to \infty }{lim}{\left(\frac{n+1}{n}\right)}^n$

An quicker way of calculating $\text{e}$ is to use the (infinite) series:

$\text{e}=1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\&ctdot;+\frac{1}{n!}+\dots$

where, we remember,

$n!=n\times \left(n-1\right)\times \left(n-2\right)\times \dots \left(3\right)\times \left(2\right)\times \left(1\right)$

(This is discussed more fully in HELM booklet  16: Sequences and Series.)

Although all functions of the form $a^x$ are called exponential functions we usually refer to ${\text{e}}^x$ as the exponential function.

Figure 2 :

Exponential functions (and variants) appear in various areas of mathematics and engineering. For example, the shape of a hanging chain or rope, under the effect of gravity, is well described by a combination of the exponential curves ${\text{e}}^{kx}$ , ${\text{e}}^{-kx}$ . The function ${\text{e}}^{-x^2}$ plays a major role in statistics; it being fundamental in the important normal distribution which describes the variability in many naturally occurring phenomena. The exponential function ${\text{e}}^{-kx}$ appears directly, again in the area of statistics, in the Poisson distribution which (amongst other things) is used to predict the number of events (which occur randomly) in a given time interval.

From now on, when we refer to an exponential function, it will be to the function ${\text{e}}^x$ (Figure 2) that we mean, unless stated otherwise.

Task!

Use a calculator to determine the following values correct to 2 d.p.

1. ${\text{e}}^{1.5}$ ,
2. ${\text{e}}^{-2}$ ,
3. ${\text{e}}^{17}$ .

Answer

Task!

Simplify the expression $\frac{{\text{e}}^{2.7}{\text{e}}^{-3\left(1.2\right)}}{{\text{e}}^2}$ and determine its numerical value to 3 d.p.

First simplify the expression:

Answer

$\frac{{\text{e}}^{2.7}{\text{e}}^{-3\left(1.2\right)}}{{\text{e}}^2}={\text{e}}^{2.7}{\text{e}}^{-3.6}{\text{e}}^{-2}={\text{e}}^{2.7-3.6-2}={\text{e}}^{-2.9}$ Now evaluate its value to 3 d.p.:

Answer

$0.055$