Title: Default for 6.1 The Exponential Function
Course: HELM Workbooks 2023-2024

Exponents revisited

Recall the meaning fo the expression  \(a^p\) where \(a\) is a positive number. We remind the reader that `\(a\)' is called the base and `\(p\)' is called the exponent (or power or index). There are various cases to consider:

If \(m,n\) are positive integers

- \(a^n=a\times a\times\dots\times a\).

- \(a^{1/n}\) means the \(n^{th}\) root of \(a\). That is, \(a^{1/n}\) is that positive number which satisfies \[\qquad (a^{1/n})\times (a^{1/n})\times\dots\times (a^{1/n})=a,\]where there are \(n\) terms on the left hand side.

- \(a^{m/n}=(a^{1/n})\times (a^{1/n})\times\dots\times(a^{1/n}),\) where there are \(m\) terms.
- \(a^{-n}=\dfrac{1}{a^n}\).

For convenience we again list the basic laws of exponents:

Key Point

\(a^ma^n=a^{m+n},\qquad\qquad \dfrac{a^m}{a^n}=a^{m-n},\qquad\qquad (a^m)^n=a^{mn},\)

\(\qquad \qquad a^1=a \qquad \text{ and }\qquad \text{ if } a\neq 0 \qquad a^0=1.\)

Example

Simplify the expression \(\dfrac{p^{n-2}p^m}{p^3p^{2m}}.\)

Solution

First we simplify the numerator: \[\qquad p^{n-2}p^m=p^{n-2+m}.  \] Next we simplify the denominator: \[\qquad p^3p^{2m}=p^{3+2m}.  \] Now we combine them and simplify: \[\qquad \dfrac{p^{n-2}p^m}{p^3p^{2m}}=\dfrac{p^{n-2+m}}{p^{3+2m}}=p^{n-2+m}p^{-3-2m} =p^{n-2+m-3-2m} =p^{n-m-5}.  \]

Exercise

Simplify the expression \(\displaystyle{\dfrac{b^[value] \,b^[value]}{b^[value]}}.\)

We can do this in two steps.

First simplify the numerator:

\(\displaystyle{b^[value]b^[value]}=\) \(b\)^

Now include the denominator:

\(\displaystyle{\dfrac{b^[value]b^[value]}{b^[value]}}\)=\(b\)^
Feedback: We have: \(\displaystyle \dfrac{b^[value] \,b^[value]}{b^[value]} = \dfrac{b^{([value])+[value]}}{b^[value]} = \dfrac{b^[value]}{b^[value]} = b^{[value]-[value]} = b^[value].\)

Exercise

Simplify the expression \( \displaystyle{\dfrac{[value][value]}{[value]}}.\)

First simplify the numerator:

\(\displaystyle{[value][value]} \)=

Now include the denominator:

\(\displaystyle{\dfrac{[value][value]}{[value]}}\)=
Feedback: We have: \[\dfrac{[value][value]}{[value]} = \dfrac[value][value] = [value]a^{[value]-[value]} = [value].\]

Defining \(a^x\) when \(x\) is any real number

So far we have given the meaning of \(a^p\) where \(p\) is an integer or a rational number, that is, one which can be written as a quotient of integers. So, if \(p\) is rational, then \[\qquad p=\dfrac{m}{n}, \qquad \text{ where } m,n \text{ are integers}.   \]

Now consider \(x\) as a real number which cannot be written as a rational number. Two common examples of these irrational numbers are \(\sqrt{2}\) and \(\pi\). What we shall do is approximate \(x\) by a rational number by working to a fixed number of decimal places. For example if \[\qquad x=3.14159265\dots,  \] then, if we are working to 3 d.p., we would write \[\qquad x\approx 3.142,  \] and this number can certainly be expressed as a rational number: \[\qquad x\approx 3.142=\dfrac{3142}{1000},  \] so in this case \[\qquad a^x=a^{3.14159\dots}\approx a^{3.142}=a^{\dfrac{3142}{1000}}. \] and the final term: \( a^{\dfrac{3142}{1000}} \) can be determined in the usual way by  calculator. Henceforth, we shall therefore assume that the expression \(a^x\) is defined for all positive values of \(a\) and for  all real values of \(x\).

This approach will allow us, with some care, to approximate \(a^x\) for any real numbers.

Exercise

By working to \(3\) d.p. find, using your calculator, the value of \(\large{[value]^{[value]}}\).

Hint.

First, approximate the value of \(\large[value]  \):

Now determine \(\large{[value]^{[value]}}\):
Feedback: We have: \[ \quad [value]\approx3.1415927\dots * [value] =[value] \approx 1.571. \] Hence, \[ [value]^[value] \approx [value]^[value]\=[value] \text{ to 3 d.p.} \]

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 below the graphs of \((0.5)^x, (0.3)^x,1^x,2^x\) 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 \[ \qquad a^x>b^x\quad \text{if}\ x>0\qquad\qquad\text{and}\qquad\qquad a^x<b^x\quad \text{if}\ x<0 \]

[Image: Exponential function for various values of a base.]

\({\bf y=a^x}\) for various values of \({\bf a}\).

The most important and widely used exponential function has the particular base \(e= 2.7182818\dots\). 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 \(e\). For example, it can be shown that the value of \(e\) is the end-point of the sequence of numbers:

\[ \quad  \left(\dfrac{2}{1}\right)^1,\quad \left(\dfrac{3}{2}\right)^2,\quad \left(\dfrac{4}{3}\right)^3 ,\quad\dots ,\qquad \left(\dfrac{17}{16}\right)^{16},\quad\dots ,\quad \left(\dfrac{65}{64}\right)^{64}\]  which, in decimal form (each to 6 d.p.) are \[ \quad 2.000000,\quad 2.250000,\quad 2.370370,\quad\dots ,\quad 2.637929,\quad \dots , 2.697345,\dots \]

 This is a slowly converging sequence. However, it does lead to a precise definition for the value of \(e\): \[e=\lim_{n\rightarrow\infty}\left(\dfrac{n+1}{n}\right)^n.\]

An quicker way of calculating \(e\) is to use the (infinite) series: \[ \quad e=1+\dfrac{1}{1!}+\dfrac{1}{2!}+\dfrac{1}{3!}+\dfrac{1}{4!}+\dots +\dfrac{1}{n!}+\dots\]  where, we remember, \[ \quad n!=n\times (n-1)\times (n-2)\times \dots (3)\times (2)\times (1).\]

Although all functions of the form \(a^x\) are called exponential functions we usually refer to \(e^x\) as  the exponential function.

Key Point

\(e^x\) is the exponential function where \(e=2.71828\dots\)

[Image: Exponential function for the base e.]

 \({\bf y = e^x}\).

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 \(e^{kx}\), \(e^{-kx}\). The function \(e^{-x^2}\) 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 \(e^{-kx}\) 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 the function \(e^x\) (Figure above) that we mean, unless stated otherwise.

Exercise

In this question you will simplify the expression \( \dfrac{e^[value]e^{[value]([value])}}{e^[value]}\) and determine its numerical value to \(3\) d.p.

First simplify the expression giving you answer with \(1\) d.p.: ^

\( \dfrac{e^[value]e^{[value]([value])}}{e^[value]}=\)   \(e\)^

Now evaluate its value to \(3\) d.p.:
Feedback: We have: \( \displaystyle \dfrac{e^[value]e^{[value]([value])}}{e^[value]} = e^[value] e^{-[value]}=e^{[value]}\approx [value]\).

Exponential growth

If \(a>1\) then it can be shown that, no matter how large \(K\) is: \[ \quad \dfrac{a^x}{x^K}\rightarrow\infty\quad \text{as}\quad x\rightarrow\infty\] That is, if \(K\) is fixed (though chosen as large as desired) then eventually, as \(x\) increases, \(a^x\) will become larger than the value \(x^K\) provided \(a>1\). The growth of \( a^x\) as \(x\) increases is called exponential growth. This limit tells us that exponential growth is eventually (large \(x\)) faster than polynomial growth of any degree.

Exercise

A function \(f(x)\) grows exponentially and is such that \(f(0)=1\) and \(f([value])=[value]\). Find the exponential curve that fits through these points. Assume the function is \(f(x)=e^{kx}\) where \(k\) is to be determined from the given information. Find the value of \(k\) up to 3 d.p.

Hint: First try values \( [value],[value], [value] \)

\(k = \)

As we have noted, the behaviour of \(e^x\) as \(x\rightarrow\infty\) is called exponential growth. In a similar manner we characterise the behaviour of the function \(e^{-x}\) as \(x\rightarrow\infty\) as exponential decay. The graphs of \(e^x\) and \(e^{-x}\) are shown in Figure 3.

FIGURE 3 \(y=\e^x\) and \(y=\e^{-x}\)

Exponential growth is very rapid and similarly exponential decay is also very rapid. In fact \(e^{-x}\) tends to zero so quickly as \(x\rightarrow\infty\) that, no matter how large the constant \(K\) is, \[ \quad x^K\e^{-x}\rightarrow 0\quad \text{as}\quad x\rightarrow\infty \]  The next Task investigates this.

Exercise

As we have noted, the behaviour of \(e^x\) as \(x\rightarrow\infty\) is called exponential growth. In a similar manner we characterise the behaviour of the function \(e^{-x}\) as \(x\rightarrow\infty\) as exponential decay. The graphs of \(e^x\) and \(e^{-x}\) are shown in Figure below.

[Image: Exponential function and its inverse for the base e.]

 \({\bf y=e^x}\) and \({\bf y=e^{-x}}\)

Exponential growth is very rapid and similarly exponential decay is also very rapid. In fact \(e^{-x}\) tends to zero so quickly as \(x\rightarrow\infty\) that, no matter how large the constant \(K\) is, \[ \quad  x^Ke^{-x}\rightarrow 0\quad \text{as}\quad x\rightarrow\infty.\]  The next Task investigates this.

Exercise

 Find, to 3 d.p., the values of

1. \( [value]^[value]\)
2. \( [value]^[value]\)
3. \(\displaystyle{[value]^[value]}\)
4. \( [value]^[value]\)
5. \( \displaystyle{[value]^[value]}\)
6. \(\pi^\pi\)
7. \(e^{[value]\pi}\)
8. \( [value]^{- [value]}\)

Plot \( y=x^3 \) and \( y=e^x\) for \( 0<x<7. \) For which integer values of \( x\) is \( e^x> x^3? \)
Feedback: For integer values of \( x, \) \( e^x> x^3 \) if \( x\geq 5. \)

Exercise

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

1. \(e^[value]=\)
2. \(e^[value]=\)
3. \(e^[value]=\)
Feedback: We have: 1. \( e^[value]=[value] \) 2. \( e^[value]=[value] \) 3. \( e^[value]=[value] \)

Exercise

Choose \(K=[value]\) in the expression \(x^[value]e^{-x}\) and calculate \(x^[value]e^{-x}\) to \(3\) d.p using your calculator for \(x=5,\ 10,\ 15,\ 20,\ 25,\ 30,\ 35\).

\(x\) | 5 | 10 | 15 | 20 | 25 | 30 | 35
\(x^[value]e^{-x}\) |  |  |  |  |  |  |
Feedback: We have \(x\) | 5 | 10 | 15 | 20 | 25 | 30 | 35 \(x^[value]e^{-x}\) | \([value] \) | \([value] \) | \([value] \) | \([value] \) | \([value] \) | \([value] \) | \([value] \)