copyright © 2015, 2008, 2011 pearson education, inc. section 4.1, slide 1 chapter 4 exponential...
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Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 1
Chapter 4Exponential Functions
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 2
4.1 Properties of Exponents
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 3
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
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 4
Properties of Exponents
If m and n are counting numbers, then
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 5
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.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 6
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.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 7
Solution
2. Write bmbn without exponents:
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 8
Solution
3. Write where c ≠ 0, without exponents:,n
bc
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 9
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.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 10
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
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 11
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
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 12
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
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 13
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
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 14
Zero Exponent
Definition
For b ≠ 0,
b0 = 1
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 15
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
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 16
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.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 17
Example: Simplifying Expressions Involving Exponents
Simplify.
1. 9b-7 2. 3. 3-1 + 4-1 3
5b
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 18
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
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 19
Properties of Integer Exponents
If m and n are integers, b ≠ 0, and c ≠ 0, then
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 20
Example: Simplifying Expressions Involving Exponents
Simplify.
1. 2.
25
32 2
3
2
bc
b c
44 7
3 2
186
b cb c
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 21
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
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 22
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
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 23
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.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 24
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)
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 25
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
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 26
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.
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 27
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
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 28
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
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 29
Example: Converting to Standard Decimal Notation
Write the number in standard decimal notation.
1. 2.53.462 10 47.38 10
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 30
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:
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 31
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
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 32
Example: Converting to Scientific Notation
Write the number in scientific notation.
1. 6,257,000,000 2. 0.00000721
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 33
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
Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.1, Slide 34
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