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ection 10.1 The Algebra of Functions

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Page 1: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Section 10.1

The Algebra of Functions

Page 2: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Section 10.1Exercise #1

Chapter 10

Page 3: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

2Let = + 16 and = 4 Find the f– . ollowing.f x x g x x

a. f + g x

b. f – g x

c. fg x

d.

fx

g

Page 4: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

a. f + g x

2 = + 16 + 4 – x x

= x2 – x + 20

2Let = + 16 and = 4 Find the f– . ollowing.f x x g x x

Page 5: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

2 = + 16 – 4 – x x

= x2 + 16 – 4 + x

b. f – g x

= x2 + x + 12

2Let = + 16 and = 4 Find the f– . ollowing.f x x g x x

Page 6: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

2 = + 16 4 – x x

= 4x2 – x3 + 64 – 16x

= – x3 + 4x2 – 16x + 64

c. fg x

2Let = + 16 and = 4 Find the f– . ollowing.f x x g x x

Page 7: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

d.

fx

g

=

x2 + 164 – x

, x 4

2Let = + 16 and = 4 Find the f– . ollowing.f x x g x x

Page 8: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Section 10.1Exercise #3

Chapter 10

Page 9: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

2If = + 2 and = + 3, find:f x x g x x

–. 2a g f

b. f g x

c. g f x

Page 10: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

–. 2a g f

= – 2g f

2– 2 = – 2 + = 62 f

– 2 6= g f g = 6 + 3 = 9

2If = + 2 and = + 3, find:f x x g x x

Page 11: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

= + 3f g x f x

2 = + 3 + 2x

= x2 + 6x + 9 + 2

= x2 + 6x + 11

b. f g x

2If = + 2 and = + 3, find:f x x g x x

Page 12: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

2 = + 2g f x g x

2 = + 2 + 3x

= x2 + 5

c. g f x

2If = + 2 and = + 3, find:f x x g x x

Page 13: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Section 10.1Exercise #4

Chapter 10

Page 14: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

fin– 1 3 + 1

If = and = ,5 2 –

d the domain 2

of

+ , – and .

xf x g x

x xf g f g fg

– 1 3 + 1 + = +

5a.

2 – 2 x

f g xx x

2 – 2 0x

2 2x

1x

D = 0, 1, is a real number x x x x

5 0x

0x

Page 15: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

– 1 3 + 1 – = –

5b.

2 – 2 x

f g xx x

D = 0, 1, is a real number x x x x

fin– 1 3 + 1

If = and = ,5 2 –

d the domain 2

of

+ , – and .

xf x g x

x xf g f g fg

Page 16: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

– 1 3 + 1

= – 2

c5

.2

xfg x

x x

D = 0, 1, is a real number x x x x

fin– 1 3 + 1

If = and = ,5 2 –

d the domain 2

of

+ , – and .

xf x g x

x xf g f g fg

Page 17: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

OBJECTIVES

A Find the sum, difference, product, and quotient of two functions.

Page 18: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

OBJECTIVES

B Find the composite of two functions.

Page 19: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

OBJECTIVES

C Find the domain of (ƒ + g)(x), (ƒ – g)(x), (ƒg)(x), and

(ƒg)(x).

Page 20: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

OBJECTIVES

D Solve an application.

Page 21: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

DEFINITIONOPERATIONS WITH FUNCTIONS

0,ƒ(x)g(x)

ƒg

(ƒ + g)(x) = ƒ(x) + g(x)

(ƒ – g)(x) = ƒ(x) – g(x)

(ƒg)(x) = ƒ(x) g(x)

(x) = g(x) ( )

Page 22: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

DEFINITIONCOMPOSITE FUNCTION

(ƒ o g)(x) = ƒ(g(x))

If ƒ and g are functions:

Page 23: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Section 10.2

Inverse Functions

Page 24: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

OBJECTIVES

A Find the inverse of a function when the function is given as a set of ordered pairs.

Page 25: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

OBJECTIVES

B Find the equation of the inverse of a function.

Page 26: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

OBJECTIVES

C Graph a function and its inverse and determine whether the inverse is a function.

Page 27: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

OBJECTIVES

D Solve applications involving functions.

Page 28: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

DEFINITION

The relation obtained by reversing the order of x and y.

INVERSE OF A FUNCTION

Page 29: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

FINDING THE EQUATION OF AN INVERSE FUNCTION

PROCEDURE

1.Interchange the roles of x and y.

2.Solve for y.

Page 30: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

DEFINITION

If y = ƒ(x) is one-to-one, the inverse of ƒ is also a function, denoted by y = ƒ –1(x).

Page 31: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Section 10.2Exercise #6

Chapter 10

Page 32: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Let = 3,5 , 5,7 and , 7, :9 findS

a. The domain and range of S

b. S–1

c. The domain and range of S–1

d. The graph of S and S–1

Page 33: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Domain of = 3,5,7S

a. The domain and range of S

Range of = 5,7,9S

b. S–1

–1 = 5,3 , 7,5 , 9,7S

Let = 3,5 , 5,7 and , 7, :9 findS

Page 34: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

c. The domain and range of S–1

–1Range of = 3,5,7S

–1Domain of = 5,7,9S

Let = 3,5 , 5,7 and , 7, :9 findS

Page 35: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

d. The graph of S and S–1

y

x 10 5 0 0

5

10

S–1

S

Let = 3,5 , 5,7 and , 7, :9 findS

Page 36: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Section 10.2Exercise #8

Chapter 10

Page 37: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

2 = = 3Find the inverse of . Is the inverse a function?f x y x

– 1 2: = 3f x x y

x3

= y2

y2 =

x3

y = ±

x3

= ± 3x3

The inverse is not a function.

Page 38: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Section 10.3

Exponential Functions

Page 39: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

OBJECTIVES

A Graph exponential functions of the form ax or a –x (a > 0).

Page 40: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

OBJECTIVES

B Determine whether an exponential function is increasing or decreasing.

Page 41: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

OBJECTIVES

C Solve applications involving exponential functions.

Page 42: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

DEFINITIONEXPONENTIAL FUNCTION

( ) ( 0 1)xƒ x = b b > , b

A function defined for all real values of x by:

Page 43: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

DEFINITION

Increasing: rises left to right.

Decreasing: falls left to right.

INCREASING AND DECREASING FUNCTIONS

Page 44: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

DEFINITIONNATURAL EXPONENTIAL FUNCTION, BASE e

ƒ(x) = ex

has the approximate value 2.178e

Page 45: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Section 10.3Exercise #9

Chapter 10

Page 46: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Graph y = 3x .

a. Is the inverse a function?

b. Is y = 3x increasing or decreasing?

Page 47: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Graph y = 3x .

a. Is the inverse a function?

x – 5 0 5

10 f x

Yesx y

0 1

1 3 –1

13

Page 48: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Graph y = 3x .

x – 5 0 5

10 f x

b. Is y = 3x increasing or decreasing?

increasing

Page 49: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Section 10.3Exercise #10

Chapter 10

Page 50: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

A radioactive substance decays so that thenumber of grams present after t years is

G = 1000e–1.4x

Find, to the nearest gram, the amount of thesubstance present,

a. At the start. b. In 2 years.

Page 51: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

When t = 0

a. At the start.

G = 1000e0

G = 1000

1000 grams are present at the start.

Page 52: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

b. In 2 years.

When t = 2

G = 1000e 1.4 2 = 1000e–2.8

G = 1000 0.06081 = 60.81 = 61

61 grams are present in 2 years.

Page 53: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Section 10.4

Logarithmic Functions and their Properties

Page 54: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

OBJECTIVES

A Graph logarithmic functions.

Page 55: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

OBJECTIVES

B Write an exponential equation in logarithmic form and a logarithmic equation in exponential form.

Page 56: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

OBJECTIVES

C Solve logarithmic equations.

Page 57: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

OBJECTIVES

D Use the properties of logarithms to simplify logarithms of products, quotients, and powers.

Page 58: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

OBJECTIVES

E Solve applications involving logarithmic functions.

Page 59: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

DEFINITION

Means the exponent to which we raise 3 to get x.

LOG3x

Page 60: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

DEFINITIONLOGARITHMIC FUNCTION

ƒ(x) = y = logbx

is equivalent to:

by = x (b > 0, b ≠ 1, and x > 0)

Page 61: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

DEFINITIONEQUIVALENCE PROPERTY

For any b > 0, b ≠ 1,

bx = by

is equivalent to x = y.

Page 62: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

DEFINITIONPROPERTIES OF LOGARITHMS

logbMN = logbM + logbN

logbMN

= logbM – logbN

logbMr = r logbM

Page 63: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

DEFINITIONOTHER PROPERTIES OF LOGARITHMS

logb 1 = 0

logb b = 1

logb bx = x

Page 64: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Section 10.4Exercise #11

Chapter 10

Page 65: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Graph on the same coordinate axes.

a. f x = 2x

b. f x = log2x

Page 66: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

x – 5 5

y

– 5

5

Graph on the same coordinate axes.

y = 2x

x y

0 1

1 2 –1 0.5

a 2. = xf x Let = y f x

Page 67: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

x – 5 5

y

– 5

5

Graph on the same coordinate axes.

y = 2x

y = log2

x

x y

1 0

2 1 0.5 –1

2 = lb. ogf x x Let = y f x

Page 68: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Section 10.4Exercise #12

Chapter 10

Page 69: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Write the equation

a. 27 = 3x in logarithmic form.

b. log5 25 = x in exponential form.

Page 70: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Write the equation

a. 27 = 3x in logarithmic form.

log3 27 = x

b. log5 25 = x in exponential form.

5x = 25

Page 71: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Section 10.4Exercise #13

Chapter 10

Page 72: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Solve.

a. log4 x = –1

b. logx 16 = 2

x = 4–1

x =

14

x 2 = 16

x = ± 16

x = 4

Page 73: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Section 10.5

Common and Natural Logarithms

Page 74: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

OBJECTIVES

A Find logarithms and their inverses base 10.

Page 75: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

OBJECTIVES

B Find logarithms and their inverses base e.

Page 76: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

OBJECTIVES

C Change the base of a logarithm.

Page 77: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

OBJECTIVES

D Graph exponential and logarithmic functions base e.

Page 78: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

OBJECTIVES

E Solve applications involving common and natural logarithms.

Page 79: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

DEFINITIONNATURAL LOGARITHMIC FUNCTION

ƒ(x) = ln x, where x means loge x and x > 0

Page 80: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

FORMULACHANGE-OF-BASE

logbM = logaMlogab

Page 81: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Section 10.5Exercise #18

Chapter 10

Page 82: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Use the change of base formula to fill in the blank.

log310 = _______

log310 =

log 10

log 3

Page 83: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

=

10.4771

= 2.0959

log310 =

log 10

log 3

Use the result part a to find a numerical approximationfor log310.

Page 84: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Section 10.5Exercise #19

Chapter 10

Page 85: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

x – 5 5

y

– 5

5

Graph.

12

xy = e

12a. =

xf x e

x y

0 1

1 1.6 –1 0.6 2 2.7

Page 86: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Section 10.5Exercise #20

Chapter 10

Page 87: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

x – 5 5

y

– 5

5

ln + 1y = x

Graph.

= ln 1. + a f x x

x y

0 0

1 0.7 2 1.1

– 0.5 – 0.7

Page 88: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Graph.

= ln b. + 1g x x

x – 5 5

y

– 5

5 y = ln x + 1 x y

1 1

2 1.7 3 2.1

0.3 0.5

Page 89: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Section 10.5Exercise #21

Chapter 10

Page 90: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Solve.

x =

12

a. 52x + 1 = 25

52x + 1 = 52

2x + 1 = 2

2x = 1

If bm = bn , then m = n

Page 91: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

x = 1

b. 3x + 1 = 92x – 1

x + 1 = 4x – 2

–3x = – 3

2 – 1 + 1 23 = 3xx

3x + 1 = 3 4x – 2

Solve.

Page 92: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Section 10.5Exercise #22

Chapter 10

Page 93: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Solve.

a. 3x = 2

x =

log 2

log 3

x =

0.30100.4771

x log 3 = log 2

x = 0.6309

Page 94: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Solve.

b. 50 = e0.20k

k = 19.56

ln 50 = 0.20k • ln e

0.39120 = 0.20k

ln 50 = ln e0.20k

Page 95: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Section 10.5Exercise #23

Chapter 10

Page 96: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Solve.

log + 2 + log – 7 1a. = x x

log + 2 – 7 = 1 x x

1 + 2 – 7 = 10x x

x2 – 5x – 14 = 10

x2 – 5x – 24 = 0

Page 97: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

x – 8 = 0

x2 – 5x – 24 = 0

– 8 + 3 = 0x x

x + 3 = 0

x = 8 x = – 3

or

= – 3 causes log –3 + 2 = log –1–3, = 8

xx x

Solve.

log + 2 + log – 7 1a. = x x

Page 98: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Solve.

3 3 log + 5 – log – 1b. = 1x x

3 log + 5 ÷ – 1 = 1 x x

1 + 5 = 3

– 1

x

x

+ 5 = 3 – 1x x

x + 5 = 3x – 3

Page 99: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

x + 5 = 3x – 3

8 = 2x

4 = x

x = 4

Solve.

b. log3 x + 5 – log3 x – 1 = 1

Page 100: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Section 10.5Exercise #24

Chapter 10

Page 101: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

The compound amount with continuous

compounding is given by A= Pert , whereP is the principal, r is the interest rate, andt is the time in years. If the rate is 8%, findhow long it takes for the money to double--for A to equal 2P (ln 2 = 0.69315.)

Page 102: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

A = 2P , r = 0.08 find t.

2P = Pe0.08t

A = Pert

2PP

= e0.08t

2 = e0.08t

Page 103: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

2 = e0.08t

ln2 = 0.08 lnt e

0.69315 = 0.08t

8.66 = t

It takes 8.66 years to double the money.

Page 104: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Section 10.5Exercise #25

Chapter 10

Page 105: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

A radioactive substance decays sothat the amount A percent at time

t (years) is A = A0e–0.5t . Find thehalf-life (time for half to decay) ofthis substance (ln 2 = 0.69315.)

Page 106: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

A =

12

A0 , find t.

A = A0e–0.5t

12

A0 = A0e–0.5t

12

= e–0.5t

2–1 = e–0.5t

Page 107: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

–1 ln 2 = – 0.5 lnt e

2–1 = e–0.5t

–0.69315 = – 0.5t

1.3863 = – 0.5t

1.386 years is the half-life of this substance.

Page 108: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

Section 10.6

Exponential and Logarithmic Equations and Applications

Page 109: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

OBJECTIVES

A Solve exponential equations.

Page 110: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

OBJECTIVES

B Solve logarithmic equations.

Page 111: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

OBJECTIVES

C Solve applications involving exponential or logarithmic equations.

Page 112: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

DEFINITION

An equation in which the variable occurs in an exponent.

EXPONENTIAL EQUATION

Page 113: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

DEFINITIONEQUIVALENCE PROPERTY

For any b > 0, b ≠ 1, bx = by

is equivalent to x = y.

Page 114: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

DEFINITIONEQUIVALENCE PROPERTY FOR LOGARITHMS

logbM = logbN

is equivalent to

M = N

Page 115: Section 10.1 The Algebra of Functions. Section 10.1 Exercise #1 Chapter 10

SOLVING LOGARITHMIC EQUATIONSPROCEDURE

1. Write equation: logbM = N2. Write equivalent exponential

equation. Solve.3. Check answer and discard

values for M ≤ 0.