1. Chapter 3: Exponential, Logistic, and Logarithmic Functions 3
Chapter 3: Exponential, Logistic, and Logarithmic Functions 3.1a &b Homework: p odd

2. 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…

3. Consider these problems:
Evaluate the expression without using a calculator. 1. 2. 3. 4.

4. We begin with an introduction to exponential functions:
First, consider:
Now, what happens
when we switch
the base and the
exponent ???
This is a familiar monomial,
and a power function…
one of the “twelve basics?”
This is an example of
an exponential function

5. 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
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!!!

6. 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. 1. 4.
Initial Value = 1, Base = 3
Initial Value = 7, Base = 1/2 2.
Nope!  g is a power func.! 5. 3.
Nope!  q is a const. func.!
Initial Value = –2, Base = 1.5

7. More Practice with Exponents
Given
, find an exact value for: 1. 2. 3. 4. 5.

8. Finding an Exponential Function from its Table of Values
Determine the formula for
the exp. func. g:
General Form: x
g(x) –2
4/9
x 3
Initial Value: –1
4/3
x 3 4
x 3 1 12
Solve for b:
x 3 2 36
The Pattern?
Final Answer:

9. Finding an Exponential Function from its Table of Values
Determine the formula for
the exp. func. h:
General Form: x
h(x)
Initial Value: –2
128
x 1/4
Solve for b: –1 32
x 1/4 8
x 1/4 1 2
Final Answer:
x 1/4 2
1/2
The Pattern?

10. How an Exponential Function Changes (a recursive formula)
For any exponential function and any real
number 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?

11. Graphs of Exponential Functions

12. We start with an “Exploration”
Graph the four given functions in the same viewing
window: [–2, 2] by [–1, 6]. What point is common to
all four graphs?
Graph the four given functions in the same viewing
window: [–2, 2] by [–1, 6]. What point is common to
all four graphs?

13. We start with an “Exploration”
Now, can we analyze these graphs???

14. Exponential Functions f(x) = b
Domain:
Continuity:
Continuous
Range:
Symmetry:
None
Boundedness:
Below by y = 0
Extrema:
None
H.A.:
y = 0
V.A.:
None
If b > 1, then also
If 0 < b < 1, then also
f is an increasing func.,
f is a decreasing func.,

15. Definition: The Natural Base e
In Sec. 1.3, we first saw the “The Exponential Function”:
Natural
(we now know that it is an exponential growth function  why?)
But what exactly is this number “e”???
Definition: The Natural Base e

16. Analysis of the Natural Exponential Function
The graph:
Domain: All reals
Range:
Continuous
Increasing for all x
No symmetry
Bounded below by y = 0
No local extrema
H.A.: y = 0
V.A.: None
End behavior:

17. Guided Practice
Describe how to transform the graph of f into the graph of g. 1.
Trans. right 1 2.
Reflect across y-axis 3.
Horizon. shrink by 1/2 4.
Reflect across both axes,
Trans. right 2 5.
Reflect across y-axis,
Vert. stretch by 5,
Trans. up 2

18. Guided Practice
Determine a formula for the exponential function whose graph
is shown.

19. Whiteboard… Exponential Decay Exponential Growth
State whether the given function is exp. growth or exp. decay,
and describe its end behavior using limits.
Exponential Decay
Exponential Growth

20. Whiteboard… x > 0 x > 0 Solve the given inequality graphically.
The graph?
x > 0
The graph?
x > 0