Chapter 3: Chapter 3: Exponential, Exponential, Logistic, and Logistic, and Logarithmic Logarithmic FunctionsFunctions

Overview of Chapter 3Overview of Chapter 3So far in this course, we have mostly studied So far in this course, we have mostly studied algebraicalgebraicfunctionsfunctions, such as , such as polyspolys, , rationalsrationals, and , and power functionspower functionsw/ rat’l exponentsw/ rat’l exponents……

The three types of functions in this chapter (exponential,The three types of functions in this chapter (exponential,logistic, and logarithmic) are called logistic, and logarithmic) are called transcendentaltranscendentalfunctionsfunctions, because they “go beyond” the basic algebra, because they “go beyond” the basic algebraoperations involved in the aforementioned functions…operations involved in the aforementioned functions…

Consider these problems:Evaluate the expression without using a calculator.

1. 3 216 33 6

2. 3125

8

3

3

125

8

5

2

6

3.2 327 2 333 23 9

4. 5 24 5 222 52 32

We begin with an introduction to exponential functions:

2f x xFirst, consider:

This is a familiar monomial,and a power function…

one of the “twelve basics?” 2xg x

Now, what happenswhen we switchthe base and the

exponent ???

This is an example ofan exponential function

Definition: Exponential Functions

xf x a b

Let a and b be real number constants. An exponential functionin x is a function that can be written in the form

where a is nonzero, b is positive, and b = 1. The constant a isthe initial value of f (the value at x = 0), and b is the base.

Note: Exponential functions are defined and continuous for allreal numbers!!!

Identifying Exponential Functions

3xf x

Which of the following are exponential functions? For those thatare exponential functions, state the initial value and the base.For those that are not, explain why not.

1.

Initial Value = 1, Base = 3Initial Value = 1, Base = 3

46g x x2.

Nope! Nope! gg is a power func.! is a power func.!

2 1.5xh x 3.

Initial Value = –2, Base = 1.5Initial Value = –2, Base = 1.5

7 2 xk x 4.

Initial Value = 7, Base = 1/2Initial Value = 7, Base = 1/2

5 6q x 5.

Nope! Nope! qq is a const. func.! is a const. func.!

More Practice with Exponents 2xf x Given

4f1.42 16

, find an exact value for:

0f2.02 1

3f 3.32

3

1 10.125

2 8

1

2f

4.1 22 2 1.414

3

2f

5.3 22 3 2

1

2

1

8 2

4

Determine the formula forthe exp. func. g:

Finding an Exponential Function from its Table of Values

x g(x)

–2 4/9

–1 4/3

0 4

1 12

2 36

xg x a b

The Pattern?The Pattern?

x 3x 3

x 3x 3

x 3x 3

x 3x 3

General Form:

0 4 4g a Initial Value:

11 4 12g b Solve for b:

3b

4 3xg x Final Answer:

Determine the formula forthe exp. func. h:

Finding an Exponential Function from its Table of Values

x h(x)

–2 128

–1 32

0 8

1 2

2 1/2

xh x a b

The Pattern?The Pattern?

x 1/4x 1/4

x 1/4x 1/4

x 1/4x 1/4

x 1/4x 1/4

General Form:

0 8 8h a Initial Value:

11 8 2h b Solve for b:

1 4b

184

x

h x

Final Answer:

How an Exponential Function Changes (a recursive formula)

xf x a b For any exponential function and any realnumber x,

If a > 0 and b > 1, the function f is increasing and is anexponential growth function. The base b is its growth factor.

1f x b f x

If a > 0 and b < 1, f is decreasing and is an exponential decayfunction. The base b is its decay factor.

Does this formula make sense with our previous examples?Does this formula make sense with our previous examples?