Skip to Content
OpenStax Logo
Buy book
  1. Preface
  2. 1 Prerequisites
    1. Introduction to Prerequisites
    2. 1.1 Real Numbers: Algebra Essentials
    3. 1.2 Exponents and Scientific Notation
    4. 1.3 Radicals and Rational Exponents
    5. 1.4 Polynomials
    6. 1.5 Factoring Polynomials
    7. 1.6 Rational Expressions
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  3. 2 Equations and Inequalities
    1. Introduction to Equations and Inequalities
    2. 2.1 The Rectangular Coordinate Systems and Graphs
    3. 2.2 Linear Equations in One Variable
    4. 2.3 Models and Applications
    5. 2.4 Complex Numbers
    6. 2.5 Quadratic Equations
    7. 2.6 Other Types of Equations
    8. 2.7 Linear Inequalities and Absolute Value Inequalities
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  4. 3 Functions
    1. Introduction to Functions
    2. 3.1 Functions and Function Notation
    3. 3.2 Domain and Range
    4. 3.3 Rates of Change and Behavior of Graphs
    5. 3.4 Composition of Functions
    6. 3.5 Transformation of Functions
    7. 3.6 Absolute Value Functions
    8. 3.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  5. 4 Linear Functions
    1. Introduction to Linear Functions
    2. 4.1 Linear Functions
    3. 4.2 Modeling with Linear Functions
    4. 4.3 Fitting Linear Models to Data
    5. Key Terms
    6. Key Concepts
    7. Review Exercises
    8. Practice Test
  6. 5 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 5.1 Quadratic Functions
    3. 5.2 Power Functions and Polynomial Functions
    4. 5.3 Graphs of Polynomial Functions
    5. 5.4 Dividing Polynomials
    6. 5.5 Zeros of Polynomial Functions
    7. 5.6 Rational Functions
    8. 5.7 Inverses and Radical Functions
    9. 5.8 Modeling Using Variation
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  7. 6 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 6.1 Exponential Functions
    3. 6.2 Graphs of Exponential Functions
    4. 6.3 Logarithmic Functions
    5. 6.4 Graphs of Logarithmic Functions
    6. 6.5 Logarithmic Properties
    7. 6.6 Exponential and Logarithmic Equations
    8. 6.7 Exponential and Logarithmic Models
    9. 6.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  8. 7 The Unit Circle: Sine and Cosine Functions
    1. Introduction to The Unit Circle: Sine and Cosine Functions
    2. 7.1 Angles
    3. 7.2 Right Triangle Trigonometry
    4. 7.3 Unit Circle
    5. 7.4 The Other Trigonometric Functions
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  9. 8 Periodic Functions
    1. Introduction to Periodic Functions
    2. 8.1 Graphs of the Sine and Cosine Functions
    3. 8.2 Graphs of the Other Trigonometric Functions
    4. 8.3 Inverse Trigonometric Functions
    5. Key Terms
    6. Key Equations
    7. Key Concepts
    8. Review Exercises
    9. Practice Test
  10. 9 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 9.1 Solving Trigonometric Equations with Identities
    3. 9.2 Sum and Difference Identities
    4. 9.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 9.4 Sum-to-Product and Product-to-Sum Formulas
    6. 9.5 Solving Trigonometric Equations
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  11. 10 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 10.1 Non-right Triangles: Law of Sines
    3. 10.2 Non-right Triangles: Law of Cosines
    4. 10.3 Polar Coordinates
    5. 10.4 Polar Coordinates: Graphs
    6. 10.5 Polar Form of Complex Numbers
    7. 10.6 Parametric Equations
    8. 10.7 Parametric Equations: Graphs
    9. 10.8 Vectors
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  12. 11 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 11.1 Systems of Linear Equations: Two Variables
    3. 11.2 Systems of Linear Equations: Three Variables
    4. 11.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 11.4 Partial Fractions
    6. 11.5 Matrices and Matrix Operations
    7. 11.6 Solving Systems with Gaussian Elimination
    8. 11.7 Solving Systems with Inverses
    9. 11.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  13. 12 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 12.1 The Ellipse
    3. 12.2 The Hyperbola
    4. 12.3 The Parabola
    5. 12.4 Rotation of Axes
    6. 12.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  14. 13 Sequences, Probability, and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 13.1 Sequences and Their Notations
    3. 13.2 Arithmetic Sequences
    4. 13.3 Geometric Sequences
    5. 13.4 Series and Their Notations
    6. 13.5 Counting Principles
    7. 13.6 Binomial Theorem
    8. 13.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  15. A | Proofs, Identities, and Toolkit Functions
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
  17. Index

Try It

6.1 Exponential Functions

1.

g(x)= 0.875 x g(x)= 0.875 x and j(x)= 1095.6 2x j(x)= 1095.6 2x represent exponential functions.

2.

5.5556 5.5556

3.

About 1.548 1.548 billion people; by the year 2031, India’s population will exceed China’s by about 0.001 billion, or 1 million people.

4.

( 0,129 ) ( 0,129 ) and ( 2,236 );N(t)=129 ( 1.3526 ) t ( 2,236 );N(t)=129 ( 1.3526 ) t

5.

f(x)=2 ( 1.5 ) x f(x)=2 ( 1.5 ) x

6.

f(x)= 2 ( 2 ) x . f(x)= 2 ( 2 ) x . Answers may vary due to round-off error. The answer should be very close to 1.4142 ( 1.4142 ) x . 1.4142 ( 1.4142 ) x .

7.

y12 1.85 x y12 1.85 x

8.

about $3,644,675.88

9.

$13,693

10.

e 0.5 0.60653 e 0.5 0.60653

11.

$3,659,823.44

12.

3.77E-26 (This is calculator notation for the number written as 3.77× 10 26 3.77× 10 26 in scientific notation. While the output of an exponential function is never zero, this number is so close to zero that for all practical purposes we can accept zero as the answer.)

6.2 Graphs of Exponential Functions

1.

The domain is ( , ); ( , ); the range is ( 0, ); ( 0, ); the horizontal asymptote is y=0. y=0.

Graph of the increasing exponential function f(x) = 4^x with labeled points at (-1, 0.25), (0, 1), and (1, 4).
2.

The domain is ( , ); ( , ); the range is ( 3, ); ( 3, ); the horizontal asymptote is y=3. y=3.

Graph of the function, f(x) = 2^(x-1)+3, with an asymptote at y=3. Labeled points in the graph are (-1, 3.25), (0, 3.5), and (1, 4).
3.

x1.608 x1.608

4.

The domain is ( , ); ( , ); the range is ( 0, ); ( 0, ); the horizontal asymptote is y=0. y=0.

Graph of the function, f(x) = (1/2)(4)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, 0.125), (0, 0.5), and (1, 2).
5.

The domain is ( , ); ( , ); the range is ( 0, ); ( 0, ); the horizontal asymptote is y=0. y=0.

Graph of the function, g(x) = -(1.25)^(-x), with an asymptote at y=0. Labeled points in the graph are (-1, 1.25), (0, 1), and (1, 0.8).
6.

f(x)= 1 3 e x 2; f(x)= 1 3 e x 2; the domain is ( , ); ( , ); the range is ( ,−2 ); ( ,−2 ); the horizontal asymptote is y=−2. y=−2.

6.3 Logarithmic Functions

1.
  1. log 10 ( 1,000,000 )=6 log 10 ( 1,000,000 )=6 is equivalent to 10 6 =1,000,000 10 6 =1,000,000
  2. log 5 ( 25 )=2 log 5 ( 25 )=2 is equivalent to 5 2 =25 5 2 =25
2.
  1. 3 2 =9 3 2 =9 is equivalent to log 3 (9)=2 log 3 (9)=2
  2. 5 3 =125 5 3 =125 is equivalent to log 5 (125)=3 log 5 (125)=3
  3. 2 1 = 1 2 2 1 = 1 2 is equivalent to log 2 ( 1 2 )=1 log 2 ( 1 2 )=1
3.

log 121 ( 11 )= 1 2 log 121 ( 11 )= 1 2 (recalling that 121 = (121) 1 2 =11 121 = (121) 1 2 =11 )

4.

log 2 ( 1 32 )=5 log 2 ( 1 32 )=5

5.

log(1,000,000)=6 log(1,000,000)=6

6.

log( 123 )2.0899 log( 123 )2.0899

7.

The difference in magnitudes was about 3.929. 3.929.

8.

It is not possible to take the logarithm of a negative number in the set of real numbers.

6.4 Graphs of Logarithmic Functions

1.

( 2, ) ( 2, )

2.

( 5, ) ( 5, )

3.
Graph of f(x)=log_(1/5)(x) with labeled points at (1/5, 1) and (1, 0). The y-axis is the asymptote.

The domain is ( 0, ), ( 0, ), the range is ( , ), ( , ), and the vertical asymptote is x=0. x=0.

4.
Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1, 0), and (3, 1).The translation function f(x)=log_3(x+4) has an asymptote at x=-4 and labeled points at (-3, 0) and (-1, 1).

The domain is ( 4, ), ( 4, ), the range ( , ), ( , ), and the asymptote x=4. x=4.

5.
Graph of two functions. The parent function is y=log_2(x), with an asymptote at x=0 and labeled points at (1, 0), and (2, 1).The translation function f(x)=log_2(x)+2 has an asymptote at x=0 and labeled points at (0.25, 0) and (0.5, 1).

The domain is ( 0, ), ( 0, ), the range is ( , ), ( , ), and the vertical asymptote is x=0. x=0.

6.
Graph of two functions. The parent function is y=log_4(x), with an asymptote at x=0 and labeled points at (1, 0), and (4, 1).The translation function f(x)=(1/2)log_4(x) has an asymptote at x=0 and labeled points at (1, 0) and (16, 1).

The domain is ( 0, ), ( 0, ), the range is ( , ), ( , ), and the vertical asymptote is x=0. x=0.

7.
Graph of f(x)=3log(x-2)+1 with an asymptote at x=2.

The domain is ( 2, ), ( 2, ), the range is ( , ), ( , ), and the vertical asymptote is x=2. x=2.

8.
Graph of f(x)=-log(-x) with an asymptote at x=0.

The domain is ( ,0 ), ( ,0 ), the range is ( , ), ( , ), and the vertical asymptote is x=0. x=0.

9.

x3.049 x3.049

10.

x=1 x=1

11.

f(x)=2ln(x+3)1 f(x)=2ln(x+3)1

6.5 Logarithmic Properties

1.

log b 2+ log b 2+ log b 2+ log b k=3 log b 2+ log b k log b 2+ log b 2+ log b 2+ log b k=3 log b 2+ log b k

2.

log 3 ( x+3 ) log 3 ( x1 ) log 3 ( x2 ) log 3 ( x+3 ) log 3 ( x1 ) log 3 ( x2 )

3.

2lnx 2lnx

4.

2ln(x) 2ln(x)

5.

log 3 16 log 3 16

6.

2logx+3logy4logz 2logx+3logy4logz

7.

2 3 lnx 2 3 lnx

8.

1 2 ln( x1 )+ln( 2x+1 )ln( x+3 )ln( x3 ) 1 2 ln( x1 )+ln( 2x+1 )ln( x+3 )ln( x3 )

9.

log( 35 46 ); log( 35 46 ); can also be written log( 5 8 ) log( 5 8 ) by reducing the fraction to lowest terms.

10.

log( 5 ( x1 ) 3 x ( 7x1 ) ) log( 5 ( x1 ) 3 x ( 7x1 ) )

11.

log x 12 ( x+5 ) 4 ( 2x+3 ) 4 ; log x 12 ( x+5 ) 4 ( 2x+3 ) 4 ; this answer could also be written log ( x 3 ( x+5 ) ( 2x+3 ) ) 4 . log ( x 3 ( x+5 ) ( 2x+3 ) ) 4 .

12.

The pH increases by about 0.301.

13.

ln8 ln0.5 ln8 ln0.5

14.

ln100 ln5 4.6051 1.6094 =2.861 ln100 ln5 4.6051 1.6094 =2.861

6.6 Exponential and Logarithmic Equations

1.

x=2 x=2

2.

x=1 x=1

3.

x= 1 2 x= 1 2

4.

The equation has no solution.

5.

x= ln3 ln( 2 3 ) x= ln3 ln( 2 3 )

6.

t=2ln( 11 3 ) t=2ln( 11 3 ) or ln ( 11 3 ) 2 ln ( 11 3 ) 2

7.

t=ln( 1 2 )= 1 2 ln( 2 ) t=ln( 1 2 )= 1 2 ln( 2 )

8.

x=ln2 x=ln2

9.

x= e 4 x= e 4

10.

x= e 5 1 x= e 5 1

11.

x9.97 x9.97

12.

x=1 x=1 or x=1 x=1

13.

t=703,800,000× ln(0.8) ln(0.5)  years  226,572,993 years. t=703,800,000× ln(0.8) ln(0.5)  years  226,572,993 years.

6.7 Exponential and Logarithmic Models

1.

f(t)=A0e0.0000000087tf(t)=A0e0.0000000087t

2.

less than 230 years, 229.3157 to be exact

3.

f(t)= A 0 e ln2 3 t f(t)= A 0 e ln2 3 t

4.

6.026 hours

5.

895 cases on day 15

6.

Exponential. y=2 e 0.5x . y=2 e 0.5x .

7.

y=3 e ( ln0.5 )x y=3 e ( ln0.5 )x

6.8 Fitting Exponential Models to Data

1.
  1. The exponential regression model that fits these data is y=522.88585984 ( 1.19645256 ) x . y=522.88585984 ( 1.19645256 ) x .
  2. If spending continues at this rate, the graduate’s credit card debt will be $4,499.38 after one year.
2.
  1. The logarithmic regression model that fits these data is y=141.91242949+10.45366573ln(x) y=141.91242949+10.45366573ln(x)
  2. If sales continue at this rate, about 171,000 games will be sold in the year 2015.
3.
  1. The logistic regression model that fits these data is y= 25.65665979 1+6.113686306 e 0.3852149008x . y= 25.65665979 1+6.113686306 e 0.3852149008x .
  2. If the population continues to grow at this rate, there will be about 25,634 25,634 seals in 2020.
  3. To the nearest whole number, the carrying capacity is 25,657.

6.1 Section Exercises

1.

Linear functions have a constant rate of change. Exponential functions increase based on a percent of the original.

3.

When interest is compounded, the percentage of interest earned to principal ends up being greater than the annual percentage rate for the investment account. Thus, the annual percentage rate does not necessarily correspond to the real interest earned, which is the very definition of nominal.

5.

exponential; the population decreases by a proportional rate. .

7.

not exponential; the charge decreases by a constant amount each visit, so the statement represents a linear function. .

9.

The forest represented by the function B(t)=82 (1.029) t . B(t)=82 (1.029) t .

11.

After t=20 t=20 years, forest A will have 43 43 more trees than forest B.

13.

Answers will vary. Sample response: For a number of years, the population of forest A will increasingly exceed forest B, but because forest B actually grows at a faster rate, the population will eventually become larger than forest A and will remain that way as long as the population growth models hold. Some factors that might influence the long-term validity of the exponential growth model are drought, an epidemic that culls the population, and other environmental and biological factors.

15.

exponential growth; The growth factor, 1.06, 1.06, is greater than 1. 1.

17.

exponential decay; The decay factor, 0.97, 0.97, is between 0 0 and 1. 1.

19.

f(x)=2000 (0.1) x f(x)=2000 (0.1) x

21.

f(x)= ( 1 6 ) 3 5 ( 1 6 ) x 5 2.93 ( 0.699 ) x f(x)= ( 1 6 ) 3 5 ( 1 6 ) x 5 2.93 ( 0.699 ) x

23.

Linear

25.

Neither

27.

Linear

29.

$10,250 $10,250

31.

$13,268.58 $13,268.58

33.

P=A(t) ( 1+ r n ) nt P=A(t) ( 1+ r n ) nt

35.

$4,572.56 $4,572.56

37.

4% 4%

39.

continuous growth; the growth rate is greater than 0. 0.

41.

continuous decay; the growth rate is less than 0. 0.

43.

$669.42 $669.42

45.

f(1)=4 f(1)=4

47.

f(1)0.2707 f(1)0.2707

49.

f(3)483.8146 f(3)483.8146

51.

y=3 5 x y=3 5 x

53.

y18 1.025 x y18 1.025 x

55.

y0.2 1.95 x y0.2 1.95 x

57.

APY= A(t)a a = a ( 1+ r 365 ) 365(1) a a = a[ ( 1+ r 365 ) 365 1 ] a = ( 1+ r 365 ) 365 1; APY= A(t)a a = a ( 1+ r 365 ) 365(1) a a = a[ ( 1+ r 365 ) 365 1 ] a = ( 1+ r 365 ) 365 1; I(n)= ( 1+ r n ) n 1 I(n)= ( 1+ r n ) n 1

59.

Let f f be the exponential decay function f(x)=a ( 1 b ) x f(x)=a ( 1 b ) x such that b>1. b>1. Then for some number n>0, n>0, f(x)=a ( 1 b ) x =a ( b 1 ) x =a ( ( e n ) 1 ) x =a ( e n ) x =a ( e ) nx . f(x)=a ( 1 b ) x =a ( b 1 ) x =a ( ( e n ) 1 ) x =a ( e n ) x =a ( e ) nx .

61.

47,622 47,622 fox

63.

1.39%; 1.39%; $155,368.09 $155,368.09

65.

$35,838.76 $35,838.76

67.

$82,247.78; $82,247.78; $449.75 $449.75

6.2 Section Exercises

1.

An asymptote is a line that the graph of a function approaches, as x x either increases or decreases without bound. The horizontal asymptote of an exponential function tells us the limit of the function’s values as the independent variable gets either extremely large or extremely small.

3.

g(x)=4 ( 3 ) x ; g(x)=4 ( 3 ) x ; y-intercept: (0,4); (0,4); Domain: all real numbers; Range: all real numbers greater than 0. 0.

5.

g(x)= 10 x +7; g(x)= 10 x +7; y-intercept: ( 0,6 ); ( 0,6 ); Domain: all real numbers; Range: all real numbers less than 7. 7.

7.

g(x)=2 ( 1 4 ) x ; g(x)=2 ( 1 4 ) x ; y-intercept: ( 0, 2 ); ( 0, 2 ); Domain: all real numbers; Range: all real numbers greater than 0. 0.

9.
Graph of two functions, g(-x)=-2(0.25)^(-x) in blue and g(x)=-2(0.25)^x in orange.

y-intercept: (0,2) (0,2)

11.
Graph of three functions, g(x)=3(2)^(x) in blue, h(x)=3(4)^(x) in green, and f(x)=3(1/4)^(x) in orange.
13.

B

15.

A

17.

E

19.

D

21.

C

23.
Graph of two functions, f(x)=(1/2)(4)^(x) in blue and -f(x)=(-1/2)(4)^x in orange.
25.
Graph of two functions, -f(x)=(4)(2)^(x)-2 in blue and f(x)=(-4)(2)^x+1 in orange.
27.
Graph of h(x)=2^(x)+3.

Horizontal asymptote: h(x)=3; h(x)=3; Domain: all real numbers; Range: all real numbers strictly greater than 3. 3.

29.

As xx, f( x ) f( x );
As xx, f( x )1 f( x )1

31.

As xx, f( x)2 f( x)2 ;
As xx, f( x ) f( x )

33.

f( x )= 4 x 3 f( x )= 4 x 3

35.

f(x)= 4 x5 f(x)= 4 x5

37.

f( x )= 4 x f( x )= 4 x

39.

y= 2 x +3 y= 2 x +3

41.

y=2 ( 3 ) x +7 y=2 ( 3 ) x +7

43.

g(6)=800+ 1 3 800.3333 g(6)=800+ 1 3 800.3333

45.

h(7)=58 h(7)=58

47.

x2.953 x2.953

49.

x0.222 x0.222

51.

The graph of G(x)= ( 1 b ) x G(x)= ( 1 b ) x is the refelction about the y-axis of the graph of F(x)= b x ; F(x)= b x ; For any real number b>0 b>0 and function f(x)= b x , f(x)= b x , the graph of ( 1 b ) x ( 1 b ) x is the the reflection about the y-axis, F(x). F(x).

53.

The graphs of g(x) g(x) and h(x) h(x) are the same and are a horizontal shift to the right of the graph of f(x); f(x); For any real number n, real number b>0, b>0, and function f(x)= b x , f(x)= b x , the graph of ( 1 b n ) b x ( 1 b n ) b x is the horizontal shift f(xn). f(xn).

6.3 Section Exercises

1.

A logarithm is an exponent. Specifically, it is the exponent to which a base b b is raised to produce a given value. In the expressions given, the base b b has the same value. The exponent, y, y, in the expression b y b y can also be written as the logarithm, log b x, log b x, and the value of x x is the result of raising b b to the power of y. y.

3.

Since the equation of a logarithm is equivalent to an exponential equation, the logarithm can be converted to the exponential equation b y =x, b y =x, and then properties of exponents can be applied to solve for x. x.

5.

The natural logarithm is a special case of the logarithm with base b b in that the natural log always has base e. e. Rather than notating the natural logarithm as log e ( x ), log e ( x ), the notation used is ln( x ). ln( x ).

7.

a c =b a c =b

9.

x y =64 x y =64

11.

15 b =a 15 b =a

13.

13 a =142 13 a =142

15.

e n =w e n =w

17.

log c (k)=d log c (k)=d

19.

log 19 y=x log 19 y=x

21.

log n ( 103 )=4 log n ( 103 )=4

23.

log y ( 39 100 )=x log y ( 39 100 )=x

25.

ln(h)=k ln(h)=k

27.

x= 2 3 = 1 8 x= 2 3 = 1 8

29.

x= 3 3 =27 x= 3 3 =27

31.

x= 9 1 2 =3 x= 9 1 2 =3

33.

x= 6 3 = 1 216 x= 6 3 = 1 216

35.

x= e 2 x= e 2

37.

32 32

39.

1.06 1.06

41.

14.125 14.125

43.

1 2 1 2

45.

4 4

47.

3 3

49.

12 12

51.

0 0

53.

10 10

55.

2.708 2.708

57.

0.151 0.151

59.

No, the function has no defined value for x=0. x=0. To verify, suppose x=0 x=0 is in the domain of the function f(x)=log(x). f(x)=log(x). Then there is some number n n such that n=log(0). n=log(0). Rewriting as an exponential equation gives: 10 n =0, 10 n =0, which is impossible since no such real number n n exists. Therefore, x=0 x=0 is not the domain of the function f(x)=log(x). f(x)=log(x).

61.

Yes. Suppose there exists a real number x x such that lnx=2. lnx=2. Rewriting as an exponential equation gives x= e 2 , x= e 2 , which is a real number. To verify, let x= e 2 . x= e 2 . Then, by definition, ln( x )=ln( e 2 )=2. ln( x )=ln( e 2 )=2.

63.

No; ln( 1 )=0, ln( 1 )=0, so ln( e 1.725 ) ln( 1 ) ln( e 1.725 ) ln( 1 ) is undefined.

65.

2 2

6.4 Section Exercises

1.

Since the functions are inverses, their graphs are mirror images about the line y=x. y=x. So for every point (a,b) (a,b) on the graph of a logarithmic function, there is a corresponding point (b,a) (b,a) on the graph of its inverse exponential function.

3.

Shifting the function right or left and reflecting the function about the y-axis will affect its domain.

5.

No. A horizontal asymptote would suggest a limit on the range, and the range of any logarithmic function in general form is all real numbers.

7.

Domain: ( , 1 2 ); ( , 1 2 ); Range: ( , ) ( , )

9.

Domain: ( 17 4 , ); ( 17 4 , ); Range: ( , ) ( , )

11.

Domain: ( 5, ); ( 5, ); Vertical asymptote: x=5 x=5

13.

Domain: ( 1 3 , ); ( 1 3 , ); Vertical asymptote: x= 1 3 x= 1 3

15.

Domain: ( 3, ); ( 3, ); Vertical asymptote: x=3 x=3

17.

Domain: ( 3 7 , )( 3 7 , );
Vertical asymptote: x= 3 7 x= 3 7 ; End behavior: as x ( 3 7 ) + ,f(x) x ( 3 7 ) + ,f(x) and as x,f(x) x,f(x)

19.

Domain: ( 3, ) ( 3, ) ; Vertical asymptote: x=3 x=3 ;
End behavior: as x 3 + x 3 + , f(x) f(x) and as xx, f(x) f(x)

21.

Domain: ( 1, ); ( 1, ); Range: ( , ); ( , ); Vertical asymptote: x=1; x=1; x-intercept: ( 5 4 ,0 ); ( 5 4 ,0 ); y-intercept: DNE

23.

Domain: ( ,0 ); ( ,0 ); Range: ( , ); ( , ); Vertical asymptote: x=0; x=0; x-intercept: ( e 2 ,0 ); ( e 2 ,0 ); y-intercept: DNE

25.

Domain: ( 0, ); ( 0, ); Range: ( , ); ( , ); Vertical asymptote: x=0; x=0; x-intercept: ( e 3 ,0 ); ( e 3 ,0 ); y-intercept: DNE

27.

B

29.

C

31.

B

33.

C

35.
Graph of two functions, g(x) = log_(1/2)(x) in orange and f(x)=log(x) in blue.
37.
Graph of two functions, g(x) = ln(1/2)(x) in orange and f(x)=e^(x) in blue.
39.

C

41.
Graph of f(x)=log_2(x+2).
43.
Graph of f(x)=ln(-x).
45.
Graph of g(x)=log(6-3x)+1.
47.

f(x)= log 2 ((x1)) f(x)= log 2 ((x1))

49.

f(x)=3 log 4 (x+2) f(x)=3 log 4 (x+2)

51.

x=2 x=2

53.

x2.303 x2.303

55.

x0.472 x0.472

57.

The graphs of f(x)= log 1 2 ( x ) f(x)= log 1 2 ( x ) and g(x)= log 2 ( x ) g(x)= log 2 ( x ) appear to be the same; Conjecture: for any positive base b1, b1, log b ( x )= log 1 b ( x ). log b ( x )= log 1 b ( x ).

59.

Recall that the argument of a logarithmic function must be positive, so we determine where x+2 x4 >0 x+2 x4 >0 . From the graph of the function f( x )= x+2 x4 , f( x )= x+2 x4 , note that the graph lies above the x-axis on the interval ( ,2 ) ( ,2 ) and again to the right of the vertical asymptote, that is ( 4, ). ( 4, ). Therefore, the domain is ( ,2 )( 4, ). ( ,2 )( 4, ).

6.5 Section Exercises

1.

Any root expression can be rewritten as an expression with a rational exponent so that the power rule can be applied, making the logarithm easier to calculate. Thus, log b ( x 1 n )= 1 n log b (x). log b ( x 1 n )= 1 n log b (x).

3.

log b ( 2 )+ log b ( 7 )+ log b ( x )+ log b ( y ) log b ( 2 )+ log b ( 7 )+ log b ( x )+ log b ( y )

5.

log b ( 13 ) log b ( 17 ) log b ( 13 ) log b ( 17 )

7.

kln(4) kln(4)

9.

ln( 7xy ) ln( 7xy )

11.

log b (4) log b (4)

13.

log b ( 7 ) log b ( 7 )

15.

15log(x)+13log(y)19log(z) 15log(x)+13log(y)19log(z)

17.

3 2 log(x)2log(y) 3 2 log(x)2log(y)

19.

8 3 log(x)+ 14 3 log(y) 8 3 log(x)+ 14 3 log(y)

21.

ln(2 x 7 ) ln(2 x 7 )

23.

log( x z 3 y ) log( x z 3 y )

25.

log 7 ( 15 )= ln( 15 ) ln( 7 ) log 7 ( 15 )= ln( 15 ) ln( 7 )

27.

log 11 ( 5 )= log 5 ( 5 ) log 5 ( 11 ) = 1 b log 11 ( 5 )= log 5 ( 5 ) log 5 ( 11 ) = 1 b

29.

log 11 ( 6 11 )= log 5 ( 6 11 ) log 5 ( 11 ) = log 5 ( 6 ) log 5 ( 11 ) log 5 ( 11 ) = ab b = a b 1 log 11 ( 6 11 )= log 5 ( 6 11 ) log 5 ( 11 ) = log 5 ( 6 ) log 5 ( 11 ) log 5 ( 11 ) = ab b = a b 1

31.

3 3

33.

2.81359 2.81359

35.

0.93913 0.93913

37.

2.23266 2.23266

39.

x=4; x=4; By the quotient rule: log 6 ( x+2 ) log 6 ( x3 )= log 6 ( x+2 x3 )=1. log 6 ( x+2 ) log 6 ( x3 )= log 6 ( x+2 x3 )=1.

Rewriting as an exponential equation and solving for x: x:

6 1 = x+2 x3 0 = x+2 x3 6 0 = x+2 x3 6( x3 ) ( x3 ) 0 = x+26x+18 x3 0 = x4 x3 x =4 6 1 = x+2 x3 0 = x+2 x3 6 0 = x+2 x3 6( x3 ) ( x3 ) 0 = x+26x+18 x3 0 = x4 x3 x =4

Checking, we find that log 6 ( 4+2 ) log 6 ( 43 )= log 6 ( 6 ) log 6 ( 1 ) log 6 ( 4+2 ) log 6 ( 43 )= log 6 ( 6 ) log 6 ( 1 ) is defined, so x=4. x=4.

41.

Let b b and n n be positive integers greater than 1. 1. Then, by the change-of-base formula, log b ( n )= log n ( n ) log n ( b ) = 1 log n ( b ) . log b ( n )= log n ( n ) log n ( b ) = 1 log n ( b ) .

6.6 Section Exercises

1.

Determine first if the equation can be rewritten so that each side uses the same base. If so, the exponents can be set equal to each other. If the equation cannot be rewritten so that each side uses the same base, then apply the logarithm to each side and use properties of logarithms to solve.

3.

The one-to-one property can be used if both sides of the equation can be rewritten as a single logarithm with the same base. If so, the arguments can be set equal to each other, and the resulting equation can be solved algebraically. The one-to-one property cannot be used when each side of the equation cannot be rewritten as a single logarithm with the same base.

5.

x= 1 3 x= 1 3

7.

n=1 n=1

9.

b= 6 5 b= 6 5

11.

x=10 x=10

13.

No solution

15.

p=log( 17 8 )7 p=log( 17 8 )7

17.

k= ln( 38 ) 3 k= ln( 38 ) 3

19.

x= ln( 38 3 )8 9 x= ln( 38 3 )8 9

21.

x=ln12  x=ln12 

23.

x= ln( 3 5 )3 8 x= ln( 3 5 )3 8

25.

no solution

27.

x=ln( 3 ) x=ln( 3 )

29.

10 2 = 1 100 10 2 = 1 100

31.

n=49 n=49

33.

k= 1 36 k= 1 36

35.

x= 9e 8 x= 9e 8

37.

n=1 n=1

39.

No solution

41.

No solution

43.

x=± 10 3 x=± 10 3

45.

x=10 x=10

47.

x=0 x=0

49.

x= 3 4 x= 3 4

51.

x=9 x=9

Graph of log_9(x)-5=y and y=-4.
53.

x= e 2 3 2.5 x= e 2 3 2.5

Graph of ln(3x)=y and y=2.
55.

x=5 x=5

Graph of log(4)+log(-5x)=y and y=2.
57.

x= e+10 4 3.2 x= e+10 4 3.2

Graph of ln(4x-10)-6=y and y=-5.
59.

No solution

Graph of log_11(-2x^2-7x)=y and y=log_11(x-2).
61.

x= 11 5 2.2 x= 11 5 2.2

Graph of log_9(3-x)=y and y=log_9(4x-8).
63.

x= 101 11 9.2 x= 101 11 9.2

Graph of 3/log_2(10)-log(x-9)=y and y=log(44).
65.

about $27,710.24 $27,710.24

Graph of f(x)=6500e^(0.0725x) with the labeled point at (20, 27710.24).
67.

about 5 years

Graph of P(t)=1650e^(0.5x) with the labeled point at (5, 20000).
69.

ln(17) 5 0.567 ln(17) 5 0.567

71.

x= log( 38 )+5log( 3 )    4log( 3 ) 2.078 x= log( 38 )+5log( 3 )    4log( 3 ) 2.078

73.

x2.2401 x2.2401

75.

x44655.7143 x44655.7143

77.

about 5.83 5.83

79.

t=ln( ( y A ) 1 k ) t=ln( ( y A ) 1 k )

81.

t=ln( ( T T s T 0 T s ) 1 k ) t=ln( ( T T s T 0 T s ) 1 k )

6.7 Section Exercises

1.

Half-life is a measure of decay and is thus associated with exponential decay models. The half-life of a substance or quantity is the amount of time it takes for half of the initial amount of that substance or quantity to decay.

3.

Doubling time is a measure of growth and is thus associated with exponential growth models. The doubling time of a substance or quantity is the amount of time it takes for the initial amount of that substance or quantity to double in size.

5.

An order of magnitude is the nearest power of ten by which a quantity exponentially grows. It is also an approximate position on a logarithmic scale; Sample response: Orders of magnitude are useful when making comparisons between numbers that differ by a great amount. For example, the mass of Saturn is 95 times greater than the mass of Earth. This is the same as saying that the mass of Saturn is about 10 2 10 2 times, or 2 orders of magnitude greater, than the mass of Earth.

7.

f(0)16.7; f(0)16.7; The amount initially present is about 16.7 units.

9.

150

11.

exponential; f(x)= 1.2 x f(x)= 1.2 x

13.

logarithmic

Graph of the question’s table.
15.

logarithmic

Graph of the question’s table.
17.
Graph of P(t)=1000/(1+9e^(-0.6t))
19.

about 1.4 1.4 years

21.

about 7.3 7.3 years

23.

4 4 half-lives; 8.18 8.18 minutes

25.

          M= 2 3 log( S S 0 ) log( S S 0 )= 3 2 M          S S 0 = 10 3M 2            S= S 0 10 3M 2           M= 2 3 log( S S 0 ) log( S S 0 )= 3 2 M          S S 0 = 10 3M 2            S= S 0 10 3M 2

27.

Let y= b x y= b x for some non-negative real number b b such that b1. b1. Then,

ln(y)=ln( b x ) ln(y)=xln(b) e ln(y) = e xln(b)       y= e xln(b) ln(y)=ln( b x ) ln(y)=xln(b) e ln(y) = e xln(b)       y= e xln(b)

29.

A=125 e ( 0.3567t ) ;A43 A=125 e ( 0.3567t ) ;A43 mg

31.

about 60 60 days

33.

f(t)=250 e (0.00914t) ; f(t)=250 e (0.00914t) ; half-life: about 76 76 minutes

35.

r0.0667, r0.0667, So the hourly decay rate is about 6.67% 6.67%

37.

f(t)=1350 e (0.03466t) ; f(t)=1350 e (0.03466t) ; after 3 hours: P(180)691,200 P(180)691,200

39.

f(t)=256 e (0.068110t) ; f(t)=256 e (0.068110t) ; doubling time: about 10 10 minutes

41.

about 88 88 minutes

43.

T(t)=90 e (0.008377t) +75, T(t)=90 e (0.008377t) +75, where t t is in minutes.

45.

about 113 113 minutes

47.

log( x )=1.5;x31.623 log( x )=1.5;x31.623

49.

MMS magnitude: 5.82 5.82

51.

N(3)71 N(3)71

53.

C

6.8 Section Exercises

1.

Logistic models are best used for situations that have limited values. For example, populations cannot grow indefinitely since resources such as food, water, and space are limited, so a logistic model best describes populations.

3.

Regression analysis is the process of finding an equation that best fits a given set of data points. To perform a regression analysis on a graphing utility, first list the given points using the STAT then EDIT menu. Next graph the scatter plot using the STAT PLOT feature. The shape of the data points on the scatter graph can help determine which regression feature to use. Once this is determined, select the appropriate regression analysis command from the STAT then CALC menu.

5.

The y-intercept on the graph of a logistic equation corresponds to the initial population for the population model.

7.

C

9.

B

11.

P(0)=22 P(0)=22 ; 175

13.

p2.67 p2.67

15.

y-intercept: ( 0,15 ) ( 0,15 )

17.

4 4 koi

19.

about 6.8 6.8 months.

21.
23.

About 38 wolves

25.

About 8.7 years

27.

f(x)= 776.682(1.426)x f(x)=776.682(1.426)x

29.
31.
33.

f(x)= 731.92e-0.3038x f(x)=731.92e-0.3038x

35.

When f(x)= 250, x3.6 f(x)=250, x3.6

37.

y=5.063+1.934log(x) y=5.063+1.934log(x)

39.
41.
43.

When f(10) 2.3 f(10)2.3

45.

When f(x)= 8, x0.82 f(x)=8, x0.82

47.

f(x)= 25.081 1+3.182 e 0.545x f(x)= 25.081 1+3.182 e 0.545x

49.

About 25

51.
53.
55.

When f(x)= 68, x4.9 f(x)=68, x4.9

57.

f(x)= 1.034341(1.281204)x f(x)=1.034341(1.281204)x ; g(x)= 4.035510 g(x)=4.035510 ; the regression curves are symmetrical about y=xy=x , so it appears that they are inverse functions.

59.

f 1 ( x ) = ln(a)-ln(cx-1) b f 1 ( x ) = ln(a)-ln(cx-1) b

Review Exercises

1.

exponential decay; The growth factor, 0.825, 0.825, is between 0 0 and 1. 1.

3.

y=0.25 ( 3 ) x y=0.25 ( 3 ) x

5.

$42,888.18 $42,888.18

7.

continuous decay; the growth rate is negative.

9.

domain: all real numbers; range: all real numbers strictly greater than zero; y-intercept: (0, 3.5);

Graph of f(x)=3.5(2^x)
11.

g(x)=7 ( 6.5 ) x ; g(x)=7 ( 6.5 ) x ; y-intercept: (0, 7); (0, 7); Domain: all real numbers; Range: all real numbers greater than 0. 0.

13.

17 x =4913 17 x =4913

15.

log a b= 2 5 log a b= 2 5

17.

x= 64 1 3 =4 x= 64 1 3 =4

19.

log( 0.000001 )=6 log( 0.000001 )=6

21.

ln( e 0.8648 )=0.8648 ln( e 0.8648 )=0.8648

23.


Graph of g(x)=log(7x+21)-4.
25.

Domain: x>5; x>5; Vertical asymptote: x=5; x=5; End behavior: as x 5 + ,f(x) x 5 + ,f(x) and as x,f(x). x,f(x).

27.

log 8 ( 65xy ) log 8 ( 65xy )

29.

ln( z xy ) ln( z xy )

31.

log y ( 12 ) log y ( 12 )

33.

ln( 2 )+ln( b )+ ln( b+1 )ln( b1 ) 2 ln( 2 )+ln( b )+ ln( b+1 )ln( b1 ) 2

35.

log 7 ( v 3 w 6 u 3 ) log 7 ( v 3 w 6 u 3 )

37.

x= log( 125 ) log( 5 ) +17 12 = 5 3 x= log( 125 ) log( 5 ) +17 12 = 5 3

39.

x=3 x=3

41.

no solution

43.

no solution

45.

x=ln( 11 ) x=ln( 11 )

47.

a= e 4 3 a= e 4 3

49.

x=± 9 5 x=± 9 5

51.

about 5.45 5.45 years

53.

f 1 ( x )= 2 4x 1 3 f 1 ( x )= 2 4x 1 3

55.

f(t)=300 ( 0.83 ) t ;f(24)3.43g f(t)=300 ( 0.83 ) t ;f(24)3.43g

57.

about 45 45 minutes

59.

about 8.5 8.5 days

61.

exponential

Graph of the table’s values.
63.

y=4 ( 0.2 ) x ; y=4 ( 0.2 ) x ; y=4 e -1.609438x y=4 e -1.609438x

65.

about 7.2 7.2 days

67.

logarithmic; y=16.687189.71860ln(x) y=16.687189.71860ln(x)

Graph of the table’s values.

Practice Test

1.

About 13 13 dolphins.

3.

$1,947 $1,947

5.

y-intercept: (0, 5) (0, 5)

Graph of f(-x)=5(0.5)^-x in blue and f(x)=5(0.5)^x in orange.
7.

8.5 a =614.125 8.5 a =614.125

9.

x= ( 1 7 ) 2 = 1 49 x= ( 1 7 ) 2 = 1 49

11.

ln( 0.716 )0.334 ln( 0.716 )0.334

13.

Domain: x<3; x<3; Vertical asymptote: x=3; x=3; End behavior: x 3 ,f(x) x 3 ,f(x) and x,f(x) x,f(x)

15.

log t ( 12 ) log t ( 12 )

17.

3ln( y )+2ln( z )+ ln( x4 ) 3 3ln( y )+2ln( z )+ ln( x4 ) 3

19.

x= ln( 1000 ) ln( 16 ) +5 3 2.497 x= ln( 1000 ) ln( 16 ) +5 3 2.497

21.

a= ln( 4 )+8 10 a= ln( 4 )+8 10

23.

no solution

25.

x=ln( 9 ) x=ln( 9 )

27.

x=± 3 3 2 x=± 3 3 2

29.

f(t)=112 e .019792t ; f(t)=112 e .019792t ; half-life: about 35 35 days

31.

T(t)=36 e 0.025131t +35;T( 60 ) 43 o F T(t)=36 e 0.025131t +35;T( 60 ) 43 o F

33.

logarithmic

Graph of the table’s values.
35.

exponential; y=15.10062 ( 1.24621 ) x y=15.10062 ( 1.24621 ) x

Graph of the table’s values.
37.

logistic; y= 18.41659 1+7.54644 e 0.68375x y= 18.41659 1+7.54644 e 0.68375x

Graph of the table’s values.
Citation/Attribution

Want to cite, share, or modify this book? This book is Creative Commons Attribution License 4.0 and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/algebra-and-trigonometry/pages/1-introduction-to-prerequisites
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/algebra-and-trigonometry/pages/1-introduction-to-prerequisites
Citation information

© Feb 7, 2020 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License 4.0 license. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.