## Exponential and Logarithmic Functions

Exponential and Logarithmic Functions S. F. Ellermeyer October 8, 1998 1. What are Exponential Functions? Let  be a positive real number. The base...
Author: Ethelbert Hood
Exponential and Logarithmic Functions S. F. Ellermeyer October 8, 1998

1.

What are Exponential Functions?

Let  be a positive real number. The base  exponential function is the function with domain  defined by     . Example 1 The base 2 exponential function is the function with domain  defined by

    .

A partial table of values for this function is as follows:

  

-3 1/8

-2 1/4

-1 1/2

-0.5  

0 1

1 2

2.5 

2 4





3 8

4 16

The graph of  is shown in Figure 1. Example 2 The base  exponential function is the function with domain  defined by

  

   

.

A partial table of values for this function is as follows:

  

-3 8

-2 4

-1 2

-0.5  

0 1

1 1/2

2 1/4

2.5 



3 1/8

4 1/16

The graph of  is shown in Figure 2. Note that the function     is really just the constant function     Hence, we will not consider the base  exponential function. Throughout our discussion of exponential functions, we will assume that we are dealing only with bases  where    and   . The following theorem summarizes some of the important properties of exponential functions. Theorem 1 (Properites of Exponential Functions) Assuming that   the following properties hold for the function     .



and 



,

1.     2.     for all    and, in fact, the range of  is  . 3. If   , then  is increasing and its graph is concave up with a horizontal asymptote at   . In particular,     as    and     as   . 1

16 14 12 10 8 6 4 2 -4

-2

00

2 x

4

Figure 1: Graph of    

16 14 12 10 8 6 4 2 -4

-2

00

2 x

Figure 2: Graph of    

2

4

4

2

-4

-2

00

2 x

4

-2

-4

Figure 3: Graphs of     and       4. If   , then  is decreasing and its graph is concave up with a horizontal asymptote at   . In particular,     as    and     as   .

2.

Inverses of Exponential Functions

Assuming that    and   , the function     is one–to–one and hence invertible. The inverse of  is called the base  logarithm function and is denoted by  . In particular, if    then    . The domain of the base  logarithm function is   and its range is . Figure 3 shows the graphs of     and      . Figure 4 shows the graphs of     and      . Exercise: Construct a partial table for     . Use graph paper to plot the points in your table and to draw the graph of     . Next, construct a corresponding partial table for      , plot the points in your table, and draw the graph of      . 3

4

2

-4

-2

00

2 x

4

-2

-4

Figure 4: Graphs of     and      

4

3.

Solving Equations Involving Exponents

The fact that the base  exponential and base  logarithm functions are inverses of each other can be used to solve equations for which the unknown occurs as part of an exponent. In particular, since     and       are inverses of each other, we know that       for all    and         for all      This means that       for all    and     for all     . Example 3 Solve the equation



  .

 Applying the function  to both sides of the equation, we obtain 

       

Thus, we have that the solution is

   .

Example 4 Solve the equation

   .

 First, we would like there to be a common base on both sides, so we recognize that    . This allows us to write the equation as

 

   

or equivalently

   .

Applying the function  to both sides, we obtain

      



which gives us Multiplying both sides by  gives or







.

    

      . The solutions of this quadratic equation are    and    Now, let us check to see if each of these is a solution of the original equation: For    we have

          

5

and 



 





    

. This shows that    is a solution to the original equation. For    we have 

      

and 

     



  

This shows that    is a solution of the original equation.

3.1

Changing Base

Note that if   and  are any positive numbers, then

    This tells us that





 

 



   



     

or (assuming that   ),

  

  

.

(1)

Equation (1) is called the Change of Base formula for logarithms. Example 5 Use the Change of Base formula to find the exact value of  .  By the Change of Base formula, we have   

4.

   





.

The ‘‘Natural’’ Exponential Function

The natural exponential function is the base exponential function where is a certain irrational number which is approximately equal to . Although it may seem strange to you to consider using base , the reason for doing it becomes clear once you have studied Calculus. It turns out that using base makes it easy to study derivatives of exponential functions - something which you will be concerned with in Calculus. The inverse of the base exponential function,     , is the base logarithm function,      . This is also called the natural logarithm function. Because use of the base logarithm is so prevalent in Mathematics, there is a special notation. Instead of writing  , we usually write . Using this notation, we obtain: 

   

 

 for all     for all     .

Since there is a key on your calculator for computing natural logarithms, you can compute a numerical value for   for any base and positive number  by using the Change of 6

Base formula. In particular, by the Change of Base formula, we have   .    

Example 6 Use your calculator to compute a numerical approximation of  



 

  

 .

Example 7 Solve the equation

  .  We apply the natural logarithm to both sides of the equation to obtain 

or

    

This gives us

5.

     

       .

More General Exponential-Type Functions

By manipulating the basic exponential functions,     , we can construct more general functions which occur frequently in applied problems. For our present purposes, we will consider the class of functions of the form

     

where and  are constants with   and   .

5.1

Functions of the Form    

Since 



  

For example, since

, the function    

 , then



 is simply the base  exponential function.

  

   

  .

If   , then    which means that  is increasing. If  , then   which  means that  is decreasing. For example, the function      is increasing and the function       is decreasing. (See Figures 5 and 6.)

5.2

Functions of the Form    

Since  is always positive,  is positive if is positive and negative if is negative. The graph of     looks similar to the graph of  itself, but it is reflected of ‘‘flipped’’ through the  axis if . The graphs in Figures 7, 8, 9, and 10 illustrate these ideas. 7

7 6 5 4 3 2 1 -4

-2

00

2 x

4



Figure 5: Graph of     

30 25 20 15 10 5 -4

-2

00

2 x

Figure 6: Graph of      

8

4

14 12 10 8 6 4 2 -4

-2

00

2 x

4



Figure 7: Graph if      

-4

-2

0

x 2

0 -2 -4 -6 -8 -10 -12 -14

Figure 8: Graph of       

9

4

60 50 40 30 20 10 -4

-2

00

2 x

4

Figure 9: Graph of       

-4

-2

0

x 2

0 -10 -20 -30 -40 -50 -60

Figure 10: Graph of       

10

4

25 20 15 10 5

-10

-5

00

5 x

10 

Figure 11: Graph of        

6.

Functions of the form      

We have seen that graphs of functions of the form     have a horizontal asymptote at   . This is because    either as    or as   . By adding a constant,  , to  , we translate the graph vertically. The vertical translation is up if    and down if  . The result is that the horizontal asymptote shifts to    . Figures 11, 12, 13, and 14 show the graphs of functions of the form       Note that each one has a horizontal asymptote at    . As we can see, the quantitative properties of a function of the form      are determined by the parameters   and  However, the qualitative behavior of members of this family of functions is essentially the same no matter what   and  are. In particular, there is a rapid increase or decrease in either the positive  or negative  direction and a ‘‘leveling off ’’ in the opposite direction. In the example which follows, we analyze a special case for the parameters   and  Following the example is an exercise which asks you to analyze the remaining cases.

Example 8 Let us analyze the properties of       in the case that   and  are all positive numbers. In particular, we would like to draw a typical graph of such a function which elucidates as many properties of the function as possible. 1. First, we note that since   , then

   as    11

10 5 -4

-2

0

2 x

4

0 -5 -10

Figure 12: Graph of         

10

5

-10

-5

00

5 x

-5

-10

Figure 13: Graph of      

12

10

14 12 10 8 6 4 2 -4 -2 0 0 2 4 6 x 8 10 12 14 -2 -4

Figure 14: Graph of       

13

Figure 15: Typical graph in the case      and    and

   as    This tells us that ‘‘blow-up’’ is in the positive  direction and ‘‘leveling off’’ is in the negative  direction. 2. Since  , the graph of  is not ‘‘flipped’’ through the  axis upon multiplication by . In particular, since  is increasing,  is also increasing. 3. Since    addition of  to  causes the graph of  to be shifted upward by  units. In particular, since  has a horizontal asymptote at    then    has a horizontal asymptote at    . 4. A typical graph of a function       where      and    is shown in Figure 15

6.1

Exercises

1. Analyze a typical function of the form       in the case where     , and  . You can do this by completing the following. (Refer to the example above for guidance.) (a) First, we note that since  , then





14

as   

and

as      This tells us that ‘‘blow-up’’ is in the ________  direction and ‘‘leveling off ’’ is in the ________  direction. (b) Since  , the graph of  (is,is not) ‘‘flipped’’ through the  axis upon multiplication by . In particular, since  is ________,  is ________. (c) Since    addition of  to  causes the graph of  to be shifted ________ by  units. In particular, since  has a horizontal asymptote at    then    has a horizontal asymptote at    . (d) A typical graph of a function       where      and   is shown below. (Supply this graph.) 2. Do the same kind of analysis for a function of the form       under each of the following conditions: – – – –

6.2

   , and   .

 ,   and  .

,    and   .

   , and  .

On the handout which was distributed in class last week, do the following exercises:

page 292 - numbers 3, 5, 11, 13, and 15-20. page 293 - number 54. page 294 - number 1 page 301 -numbers 1, 3, 5, 7, and 18. page 302 - number 28 page 303 - number 29 page 311 - every other odd-numbered problem starting with 1 and ending with 21. page 317 - numbers 45, 47, and 49.

15