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...
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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  .

Additional Homework Exercises

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.

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