Download Chapter 3: Exponential, Logistic, and Logarithmic Functions

Chapter 3:
Exponential, Logistic,
and Logarithmic
Functions
Overview of Chapter 3
So far in this course, we have mostly studied algebraic
functions, such as polys, rationals, and power functions
w/ rat’l exponents…
The three types of functions in this chapter (exponential,
logistic, and logarithmic) are called transcendental
functions, because they “go beyond” the basic algebra
operations involved in the aforementioned functions…
Consider these problems:
Evaluate the expression without using a calculator.
3
216 
 6
3
 6
3
125
125
5
2. 3

 3
8
2
8
23
3 23
2
3. 27   3   3  9
1.
4.
52
4
 2
3

2 52
 2  32
5
We begin with an introduction
to exponential functions:
First, consider:
f  x  x
2
This is a familiar monomial,
and a power function…
one of the “twelve basics?”
Now, what happens
when we switch
the base and the
exponent ???
g  x  2
x
This is an example of
an exponential function
Definition: Exponential
Functions
Let a and b be real number constants. An exponential function
in x is a function that can be written in the form
f  x  a b
x
where a is nonzero, b is positive, and b = 1. The constant a is
the initial value of f (the value at x = 0), and b is the base.
Note: Exponential functions are defined and continuous for all
real numbers!!!
Identifying Exponential Functions
Which of the following are exponential functions? For those that
are exponential functions, state the initial value and the base.
For those that are not, explain why not.
x
1.
x
4.
Initial Value = 1, Base = 3
Initial Value = 7, Base = 1/2
4
f  x  3
2.
k  x  7 2
g  x   6x
Nope!  g is a power func.!
3.
h  x   2 1.5
x
Initial Value = –2, Base = 1.5
5.
q  x  5 6

Nope!  q is a const. func.!
More Practice with Exponents
f  x   2 , find an exact value for:
4
0
2. f  0   2  1
f  4   2  16
1 1
3
f  3  2  3   0.125
2
8
1
12
f    2  2  1.414
2
1
1
2
 3
3 2
f    2  3 2 

2
8
4
 2
Given
1.
3.
4.
5.
x
Finding an Exponential Function from its Table of Values
Determine the formula for
the exp. func. g:
General Form:
x
x
g(x)
g x a b
 
–2
4/9
–1
4/3
0
4
1
2
12
36
x3
x3
x3
x3
The Pattern?
Initial Value:
g  0  4  a  4
Solve for b:
g 1  4 b  12  b  3
1
Final Answer:
g  x  4 3
x
Finding an Exponential Function from its Table of Values
Determine the formula for
the exp. func. h:
x
General Form: h x  a b
x
h(x)
Initial Value: h 0  8  a  8
–2
128
x 1/4 Solve for b:
–1
32
x 1/4 h 1  8 b1  2  b  1 4
0
8
x 1/4
Final Answer:
1
2
x
1/4
x
2
1/2
 
 

The Pattern?
1
h  x  8  
4
How an Exponential Function
Changes (a recursive formula)
For any exponential function
number x,
f  x  a b
x
and any real
f  x  1  b f  x 
If a > 0 and b > 1, the function f is increasing and is an
exponential growth function. The base b is its growth factor.
If a > 0 and b < 1, f is decreasing and is an exponential decay
function. The base b is its decay factor.
Does this formula make sense with our previous examples?