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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