Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 1 Chapter 4 Exponential...

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 1

Chapter 4Exponential Functions


4.1 Properties of Exponents


Exponent

Definition

For any counting number n,

We refer to bn as the power, the nth power of b, or b raised to the nth power. We call b the base and n the exponent.

factors of

...n

n b

b b b b b


Properties of Exponents

If m and n are counting numbers, then


Example: Meaning of Exponential Properties

1. Show that b2b3 = b5.

2. Show that bmbn = bm + n, where m and n are counting numbers

3. Show that where n is a counting,n n

n

b bc c

number and c ≠ 0.


Solution

1. By writing b2b3 without exponents, we see52 3 ( )( )b bb bbbbb bb b b b

We can verify that this result is correct for various constant bases by examining graphing calculator tables for both y = x2x3 and y = x5.


Solution

2. Write bmbn without exponents:


Solution

3. Write where c ≠ 0, without exponents:,n

bc


Simplifying Expressions Involving Exponents

An expression involving exponents is simplified if

1. It includes no parentheses.2. Each variable or constant appears as a base as few

times as possible. For example, we write x2x4 = x6.

3. Each numerical expression (such as 72) has been calculated and each numerical fraction has been simplified.

4. Each exponent is positive.


Example: Simplifying Expressions Involving Exponents

Simplify.

1. (2b2c3)5 2. (3b3c4)(2b6c2)

3. 4.7 6

2 5

312

b cb c

47 8

2 5 3

2416

b cb c d


Solution

5 52 3 53 52(2 ) 2 ( ) ( )b c b c10 1532b c

1.

2. 43 6 3 4 262(3 )(2 ) (3 2)( )( )b b b bc c c c 696b c


Solution

3. 4.6 6 5

5

7 7 2

2

312 4

b bc ccb

5

4b c

8 37 5

2

4

5 3 3

424 3

16 2d db ccb

c b

5 3 4

43

3

2

b c

d

5 34 4

43

4

4

3 ( ) ( )2 ( )b c

d

20 12

12

8116b c

d


Simplifying Expressions Involving Exponents

Warning

The expressions 3b2 and (3b)2 are not equivalent expressions:

23 3b b b

23 3 3 9 bb b b b


Zero Exponent

Definition

For b ≠ 0,

b0 = 1


Negative integer exponent

Definition

If b ≠ 0 and n is a counting number, then

b-n

In words, to find b-n, take its reciprocal and switch the sign of the exponent.

1nb


Negative Exponent in a Denominator

If b ≠ 0 and n is a counting number, then

In words, to find take its reciprocal and switch

1 nn b

b

1,nb

the sign of the exponent.



Simplify.

1. 9b-7 2. 3. 3-1 + 4-1 3

5b


Solution

1.

2.

3.

77 7

1 99 9b

b b

33 3

5 15 5b

b b

1 1 1 1 4 3 73 4

3 4 12 12 12


Properties of Integer Exponents

If m and n are integers, b ≠ 0, and c ≠ 0, then



Simplify.

1. 2.

25

32 2

3

2

bc

b c

44 7

3 2

186

b cb c


Solution

2 22 25 5

2 2 23 3 323

3 3

2 2

bc b c

b c b c 1.

102

6 6

98

bb

cc

2 ( 6 1) 0 698

cb

8 498

b c


Solution

2. 4

4

44( 3

72

2)

3718

36

cc

cb

bb

1 5 43b c

4 44 1 53 b c

4 4 203 b c 4

4 203bc

4

2081bc


Exponential function

Definition

An exponential function is a function whose equation can be put into the form

f(x) = abx

Where a ≠ 0, b > 0, and b ≠ 1. The constant b is called the base.


Example: Evaluating Exponential Functions

For f(x) = 3(2)x and g(x) = 5x, find the following.

1. f(3) 2. f(–4) 3. g(a + 3) 4. g(2a)


Solution

1.

2.

3.

4.

3( ) 3(2) 8 23 3 4f

44 3 3

( ) 3(2)2 1

46

f

3 3( ) 5 5 5 125(5)3 a aaag

2 2( ) 5 5 252 a a aag

f(x) = 3(2)x and g(x) = 5x


Exponential Functions

Warning

It is a common error to confuse exponential functions such as E(x) = 2x with linear functions such as L(x) = 2x.

For the exponential function, the variable x is the exponent.

For the linear function, the variable x is a base.


Scientific notation

Definition

A number is written in scientific notation if it has the form where k is an integers and either –10 < N ≤ –1 or 1 ≤ N < 10.

10 ,kN


Converting from Scientific Notation to Standard Decimal Notation

To write the scientific notation in standard decimal notation, we move the decimal point of the number N as follows:

• If k is positive, we multiply N by 10 k times; hence, we move the decimal point k places to the right.• If k is negative, we divide N by 10 k times; hence, we move the decimal k places to the left.

10kN


Example: Converting to Standard Decimal Notation

Write the number in standard decimal notation.

1. 2.53.462 10 47.38 10


Solution

1. We multiply 3.462 by 10 five times; hence, we move the decimal point of 3.462 five places to the right:

2. We divide 7.38 by 10 four times; hence, we move the decimal point of 7.38 four places to the left:


Converting from Standard Decimal Notation to Scientific Notation

To write a number in scientific notation, count the number of places k that the decimal point must be moved so the new number N meets the condition –10 < N ≤ –1 or 1 ≤ N < 10:

• If the decimal point is moved to the left, then the scientific notation is written as • If the decimal point is moved to the right, then the scientific notation is written as

10 .kN

10 .kN


Example: Converting to Scientific Notation

Write the number in scientific notation.

1. 6,257,000,000 2. 0.00000721


Solution

1. In scientific notation, we would have

We must move the decimal point of 6.257 nine places to the right to get 6,257,000,000. So, k = 9 and the scientific notation is

6.257 10k

96.257 10


Solution

2. In scientific notation, we would have

We must move the decimal point of 7.21 six places to the left to get 0.00000721. So, k = –6 and the scientific notation is

7.21 10k

67.21 10

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 1 Chapter 4 Exponential...

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