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Algebra and Trigonometry

6.4 Graphs of Logarithmic Functions

Algebra and Trigonometry6.4 Graphs of Logarithmic Functions
  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

Learning Objectives

In this section, you will:
  • Identify the domain of a logarithmic function.
  • Graph logarithmic functions.

In Graphs of Exponential Functions, we saw how creating a graphical representation of an exponential model gives us another layer of insight for predicting future events. How do logarithmic graphs give us insight into situations? Because every logarithmic function is the inverse function of an exponential function, we can think of every output on a logarithmic graph as the input for the corresponding inverse exponential equation. In other words, logarithms give the cause for an effect.

To illustrate, suppose we invest $2500 $2500 in an account that offers an annual interest rate of 5%, 5%, compounded continuously. We already know that the balance in our account for any year t t can be found with the equation A=2500 e 0.05t . A=2500 e 0.05t .

But what if we wanted to know the year for any balance? We would need to create a corresponding new function by interchanging the input and the output; thus we would need to create a logarithmic model for this situation. By graphing the model, we can see the output (year) for any input (account balance). For instance, what if we wanted to know how many years it would take for our initial investment to double? Figure 1 shows this point on the logarithmic graph.

A graph titled, “Logarithmic Model Showing Years as a Function of the Balance in the Account”. The x-axis is labeled, “Account Balance”, and the y-axis is labeled, “Years”. The line starts at $25,000 on the first year. The graph also notes that the balance reaches $5,000 near year 14.
Figure 1

In this section we will discuss the values for which a logarithmic function is defined, and then turn our attention to graphing the family of logarithmic functions.

Finding the Domain of a Logarithmic Function

Before working with graphs, we will take a look at the domain (the set of input values) for which the logarithmic function is defined.

Recall that the exponential function is defined as y= b x y= b x for any real number x x and constant b>0, b>0, b1, b1, where

  • The domain of y y is ( , ). ( , ).
  • The range of y y is ( 0, ). ( 0, ).

In the last section we learned that the logarithmic function y= log b ( x ) y= log b ( x ) is the inverse of the exponential function y= b x . y= b x . So, as inverse functions:

  • The domain of y= log b ( x ) y= log b ( x ) is the range of y= b x : y= b x : ( 0, ). ( 0, ).
  • The range of y= log b ( x ) y= log b ( x ) is the domain of y= b x : y= b x : ( , ). ( , ).

Transformations of the parent function y= log b ( x ) y= log b ( x ) behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations—shifts, stretches, compressions, and reflections—to the parent function without loss of shape.

In Graphs of Exponential Functions we saw that certain transformations can change the range of y= b x . y= b x . Similarly, applying transformations to the parent function y= log b ( x ) y= log b ( x ) can change the domain. When finding the domain of a logarithmic function, therefore, it is important to remember that the domain consists only of positive real numbers. That is, the argument of the logarithmic function must be greater than zero.

For example, consider f(x)= log 4 ( 2x3 ). f(x)= log 4 ( 2x3 ). This function is defined for any values of x x such that the argument, in this case 2x3, 2x3, is greater than zero. To find the domain, we set up an inequality and solve for x: x:

2x3>0 Show the argument greater than zero. 2x>3 Add 3. x>1.5 Divide by 2. 2x3>0 Show the argument greater than zero. 2x>3 Add 3. x>1.5 Divide by 2.

In interval notation, the domain of f(x)= log 4 ( 2x3 ) f(x)= log 4 ( 2x3 ) is ( 1.5, ). ( 1.5, ).

How To

Given a logarithmic function, identify the domain.

  1. Set up an inequality showing the argument greater than zero.
  2. Solve for x. x.
  3. Write the domain in interval notation.

Example 1

Identifying the Domain of a Logarithmic Shift

What is the domain of f(x)= log 2 (x+3)? f(x)= log 2 (x+3)?

Try It #1

What is the domain of f(x)= log 5 (x2)+1? f(x)= log 5 (x2)+1?

Example 2

Identifying the Domain of a Logarithmic Shift and Reflection

What is the domain of f(x)=log(52x)? f(x)=log(52x)?

Try It #2

What is the domain of f(x)=log(x5)+2? f(x)=log(x5)+2?

Graphing Logarithmic Functions

Now that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions. The family of logarithmic functions includes the parent function y= log b ( x ) y= log b ( x ) along with all its transformations: shifts, stretches, compressions, and reflections.

We begin with the parent function y= log b ( x ). y= log b ( x ). Because every logarithmic function of this form is the inverse of an exponential function with the form y= b x , y= b x , their graphs will be reflections of each other across the line y=x. y=x. To illustrate this, we can observe the relationship between the input and output values of y= 2 x y= 2 x and its equivalent x= log 2 (y) x= log 2 (y) in Table 1.

x x 3 3 2 2 1 1 0 0 1 1 2 2 3 3
2 x =y 2 x =y 1 8 1 8 1 4 1 4 1 2 1 2 1 1 2 2 4 4 8 8
log 2 ( y )=x log 2 ( y )=x 3 3 2 2 1 1 0 0 1 1 2 2 3 3
Table 1

Using the inputs and outputs from Table 1, we can build another table to observe the relationship between points on the graphs of the inverse functions f(x)= 2 x f(x)= 2 x and g(x)= log 2 (x). g(x)= log 2 (x). See Table 2.

f(x)= 2 x f(x)= 2 x ( 3, 1 8 ) ( 3, 1 8 ) ( 2, 1 4 ) ( 2, 1 4 ) ( 1, 1 2 ) ( 1, 1 2 ) ( 0,1 ) ( 0,1 ) ( 1,2 ) ( 1,2 ) ( 2,4 ) ( 2,4 ) ( 3,8 ) ( 3,8 )
g(x)= log 2 ( x ) g(x)= log 2 ( x ) ( 1 8 ,3 ) ( 1 8 ,3 ) ( 1 4 ,2 ) ( 1 4 ,2 ) ( 1 2 ,1 ) ( 1 2 ,1 ) ( 1,0 ) ( 1,0 ) ( 2,1 ) ( 2,1 ) ( 4,2 ) ( 4,2 ) ( 8,3 ) ( 8,3 )
Table 2

As we’d expect, the x- and y-coordinates are reversed for the inverse functions. Figure 2 shows the graph of f f and g. g.

Graph of two functions, f(x)=2^x and g(x)=log_2(x), with the line y=x denoting the axis of symmetry.
Figure 2 Notice that the graphs of f( x )= 2 x f( x )= 2 x and g( x )= log 2 ( x ) g( x )= log 2 ( x ) are reflections about the line y=x. y=x.

Observe the following from the graph:

  • f(x)= 2 x f(x)= 2 x has a y-intercept at (0,1) (0,1) and g(x)= log 2 ( x ) g(x)= log 2 ( x ) has an x- intercept at (1,0). (1,0).
  • The domain of f(x)= 2 x , f(x)= 2 x , ( , ), ( , ), is the same as the range of g(x)= log 2 ( x ). g(x)= log 2 ( x ).
  • The range of f(x)= 2 x , f(x)= 2 x , ( 0, ), ( 0, ), is the same as the domain of g(x)= log 2 ( x ). g(x)= log 2 ( x ).

Characteristics of the Graph of the Parent Function, f(x)= log b ( x ): f(x)= log b ( x ):

For any real number x x and constant b>0, b>0, b1, b1, we can see the following characteristics in the graph of f(x)= log b ( x ): f(x)= log b ( x ):

  • one-to-one function
  • vertical asymptote: x=0 x=0
  • domain: (0,) (0,)
  • range: ( , ) ( , )
  • x-intercept: (1,0) (1,0) and key point (b,1) (b,1)
  • y-intercept: none
  • increasing if b>1 b>1
  • decreasing if 0<b<1 0<b<1

See Figure 3.

Two graphs of the function f(x)=log_b(x) with points (1,0) and (b, 1). The first graph shows the line when b>1, and the second graph shows the line when 0<b<1.
Figure 3

Figure 4 shows how changing the base b b in f(x)= log b ( x ) f(x)= log b ( x ) can affect the graphs. Observe that the graphs compress vertically as the value of the base increases. (Note: recall that the function ln( x ) ln( x ) has base e2.718.) e2.718.)

Graph of three equations: y=log_2(x) in blue, y=ln(x) in orange, and y=log(x) in red. The y-axis is the asymptote.
Figure 4 The graphs of three logarithmic functions with different bases, all greater than 1.

How To

Given a logarithmic function with the form f(x)= log b ( x ), f(x)= log b ( x ), graph the function.

  1. Draw and label the vertical asymptote, x=0. x=0.
  2. Plot the x-intercept, ( 1,0 ). ( 1,0 ).
  3. Plot the key point ( b,1 ). ( b,1 ).
  4. Draw a smooth curve through the points.
  5. State the domain, ( 0, ), ( 0, ), the range, ( , ), ( , ), and the vertical asymptote, x=0. x=0.

Example 3

Graphing a Logarithmic Function with the Form f(x) = logb(x).

Graph f(x)= log 5 ( x ). f(x)= log 5 ( x ). State the domain, range, and asymptote.

Try It #3

Graph f(x)= log 1 5 (x). f(x)= log 1 5 (x). State the domain, range, and asymptote.

Graphing Transformations of Logarithmic Functions

As we mentioned in the beginning of the section, transformations of logarithmic graphs behave similarly to those of other parent functions. We can shift, stretch, compress, and reflect the parent function y= log b ( x ) y= log b ( x ) without loss of shape.

Graphing a Horizontal Shift of f(x) = logb(x)

When a constant c c is added to the input of the parent function f(x)=lo g b (x), f(x)=lo g b (x), the result is a horizontal shift c c units in the opposite direction of the sign on c. c. To visualize horizontal shifts, we can observe the general graph of the parent function f(x)= log b ( x ) f(x)= log b ( x ) and for c>0 c>0 alongside the shift left, g(x)= log b ( x+c ), g(x)= log b ( x+c ), and the shift right, h(x)= log b ( xc ). h(x)= log b ( xc ). See Figure 6.

Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0  and g(x)=log_b(x+c) is the translation function with an asymptote at x=-c. This shows the translation of shifting left.
Figure 6

Horizontal Shifts of the Parent Function f(x)= log b ( x ) f(x)= log b ( x )

For any constant c, c, the function f(x)= log b ( x+c ) f(x)= log b ( x+c )

  • shifts the parent function y= log b ( x ) y= log b ( x ) left c c units if c>0. c>0.
  • shifts the parent function y= log b ( x ) y= log b ( x ) right c c units if c<0. c<0.
  • has the vertical asymptote x=c. x=c.
  • has domain ( c, ). ( c, ).
  • has range ( , ). ( , ).

How To

Given a logarithmic function with the form f(x)= log b ( x+c ), f(x)= log b ( x+c ), graph the translation.

  1. Identify the horizontal shift:
    1. If c>0, c>0, shift the graph of f(x)= log b ( x ) f(x)= log b ( x ) left c c units.
    2. If c<0, c<0, shift the graph of f(x)= log b ( x ) f(x)= log b ( x ) right c c units.
  2. Draw the vertical asymptote x=c. x=c.
  3. Identify three key points from the parent function. Find new coordinates for the shifted functions by subtracting c c from the x x coordinate.
  4. Label the three points.
  5. The Domain is ( c, ), ( c, ), the range is ( , ), ( , ), and the vertical asymptote is x=c. x=c.

Example 4

Graphing a Horizontal Shift of the Parent Function y = logb(x)

Sketch the horizontal shift f(x)= log 3 (x2) f(x)= log 3 (x2) alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.

Try It #4

Sketch a graph of f(x)= log 3 (x+4) f(x)= log 3 (x+4) alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.

Graphing a Vertical Shift of y = logb(x)

When a constant d d is added to the parent function f(x)= log b ( x ), f(x)= log b ( x ), the result is a vertical shift d d units in the direction of the sign on d. d. To visualize vertical shifts, we can observe the general graph of the parent function f(x)= log b ( x ) f(x)= log b ( x ) alongside the shift up, g(x)= log b ( x )+d g(x)= log b ( x )+d and the shift down, h(x)= log b ( x )d. h(x)= log b ( x )d. See Figure 8.

Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0  and g(x)=log_b(x)+d is the translation function with an asymptote at x=0. This shows the translation of shifting up. Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0  and g(x)=log_b(x)-d is the translation function with an asymptote at x=0. This shows the translation of shifting down.
Figure 8

Vertical Shifts of the Parent Function y= log b ( x ) y= log b ( x )

For any constant d, d, the function f(x)= log b ( x )+d f(x)= log b ( x )+d

  • shifts the parent function y= log b ( x ) y= log b ( x ) up d d units if d>0. d>0.
  • shifts the parent function y= log b ( x ) y= log b ( x ) down d d units if d<0. d<0.
  • has the vertical asymptote x=0. x=0.
  • has domain ( 0, ). ( 0, ).
  • has range ( , ). ( , ).

How To

Given a logarithmic function with the form f(x)= log b ( x )+d, f(x)= log b ( x )+d, graph the translation.

  1. Identify the vertical shift:
    • If d>0, d>0, shift the graph of f(x)= log b ( x ) f(x)= log b ( x ) up d d units.
    • If d<0, d<0, shift the graph of f(x)= log b ( x ) f(x)= log b ( x ) down d d units.
  2. Draw the vertical asymptote x=0. x=0.
  3. Identify three key points from the parent function. Find new coordinates for the shifted functions by adding d d to the y y coordinate.
  4. Label the three points.
  5. The domain is ( 0, ), ( 0, ), the range is ( , ), ( , ), and the vertical asymptote is x=0. x=0.

Example 5

Graphing a Vertical Shift of the Parent Function y = logb(x)

Sketch a graph of f(x)= log 3 (x)2 f(x)= log 3 (x)2 alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Try It #5

Sketch a graph of f(x)= log 2 (x)+2 f(x)= log 2 (x)+2 alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Graphing Stretches and Compressions of y = logb(x)

When the parent function f(x)= log b ( x ) f(x)= log b ( x ) is multiplied by a constant a>0, a>0, the result is a vertical stretch or compression of the original graph. To visualize stretches and compressions, we set a>1 a>1 and observe the general graph of the parent function f(x)= log b ( x ) f(x)= log b ( x ) alongside the vertical stretch, g(x)=a log b ( x ) g(x)=a log b ( x ) and the vertical compression, h(x)= 1 a log b ( x ). h(x)= 1 a log b ( x ). See Figure 10.

Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0  and g(x)=alog_b(x) when a>1 is the translation function with an asymptote at x=0. The graph note the intersection of the two lines at (1, 0). This shows the translation of a vertical stretch.
Figure 10

Vertical Stretches and Compressions of the Parent Function y= log b ( x ) y= log b ( x )

For any constant a>1, a>1, the function f(x)=a log b ( x ) f(x)=a log b ( x )

  • stretches the parent function y= log b ( x ) y= log b ( x ) vertically by a factor of a a if a>1. a>1.
  • compresses the parent function y= log b ( x ) y= log b ( x ) vertically by a factor of a a if 0<a<1. 0<a<1.
  • has the vertical asymptote x=0. x=0.
  • has the x-intercept ( 1,0 ). ( 1,0 ).
  • has domain ( 0, ). ( 0, ).
  • has range ( , ). ( , ).

How To

Given a logarithmic function with the form f(x)=a log b ( x ), f(x)=a log b ( x ), a>0, a>0, graph the translation.

  1. Identify the vertical stretch or compressions:
    • If |a|>1, |a|>1, the graph of f(x)= log b ( x ) f(x)= log b ( x ) is stretched by a factor of a a units.
    • If |a|<1, |a|<1, the graph of f(x)= log b ( x ) f(x)= log b ( x ) is compressed by a factor of a a units.
  2. Draw the vertical asymptote x=0. x=0.
  3. Identify three key points from the parent function. Find new coordinates for the shifted functions by multiplying the y y coordinates by a. a.
  4. Label the three points.
  5. The domain is ( 0, ), ( 0, ), the range is ( , ), ( , ), and the vertical asymptote is x=0. x=0.

Example 6

Graphing a Stretch or Compression of the Parent Function y = logb(x)

Sketch a graph of f(x)=2 log 4 (x) f(x)=2 log 4 (x) alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Try It #6

Sketch a graph of f(x)= 1 2 log 4 (x) f(x)= 1 2 log 4 (x) alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Example 7

Combining a Shift and a Stretch

Sketch a graph of f(x)=5log(x+2). f(x)=5log(x+2). State the domain, range, and asymptote.

Try It #7

Sketch a graph of the function f(x)=3log(x2)+1. f(x)=3log(x2)+1. State the domain, range, and asymptote.

Graphing Reflections of f(x) = logb(x)

When the parent function f(x)= log b ( x ) f(x)= log b ( x ) is multiplied by −1, −1, the result is a reflection about the x-axis. When the input is multiplied by −1, −1, the result is a reflection about the y-axis. To visualize reflections, we restrict b>1, b>1, and observe the general graph of the parent function f(x)= log b ( x ) f(x)= log b ( x ) alongside the reflection about the x-axis, g(x)= −log b ( x ) g(x)= −log b ( x ) and the reflection about the y-axis, h(x)= log b ( x ). h(x)= log b ( x ).

Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0  and g(x)=-log_b(x) when b>1 is the translation function with an asymptote at x=0. The graph note the intersection of the two lines at (1, 0). This shows the translation of a reflection about the x-axis.
Figure 13

Reflections of the Parent Function y= log b ( x ) y= log b ( x )

The function f(x)= −log b ( x ) f(x)= −log b ( x )

  • reflects the parent function y= log b ( x ) y= log b ( x ) about the x-axis.
  • has domain, ( 0, ), ( 0, ), range, ( , ), ( , ), and vertical asymptote, x=0, x=0, which are unchanged from the parent function.


The function f(x)= log b ( x ) f(x)= log b ( x )

  • reflects the parent function y= log b ( x ) y= log b ( x ) about the y-axis.
  • has domain ( ,0 ). ( ,0 ).
  • has range, ( , ), ( , ), and vertical asymptote, x=0, x=0, which are unchanged from the parent function.

How To

Given a logarithmic function with the parent function f(x)= log b ( x ), f(x)= log b ( x ), graph a translation.

If f(x)= log b (x) If f(x)= log b (x) If f(x)= log b (x) If f(x)= log b (x)
  1. Draw the vertical asymptote, x=0. x=0.
  1. Draw the vertical asymptote, x=0. x=0.
  1. Plot the x-intercept, ( 1,0 ). ( 1,0 ).
  1. Plot the x-intercept, ( 1,0 ). ( 1,0 ).
  1. Reflect the graph of the parent function f(x)= log b ( x ) f(x)= log b ( x ) about the x-axis.
  1. Reflect the graph of the parent function f(x)= log b ( x ) f(x)= log b ( x ) about the y-axis.
  1. Draw a smooth curve through the points.
  1. Draw a smooth curve through the points.
  1. State the domain, ( 0, ), ( 0, ), the range, ( , ), ( , ), and the vertical asymptote x=0. x=0.
  1. State the domain, ( ,0 ), ( ,0 ), the range, ( , ), ( , ), and the vertical asymptote x=0. x=0.
Table 3

Example 8

Graphing a Reflection of a Logarithmic Function

Sketch a graph of f(x)=log(x) f(x)=log(x) alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Try It #8

Graph f(x)=log(x). f(x)=log(x). State the domain, range, and asymptote.

How To

Given a logarithmic equation, use a graphing calculator to approximate solutions.

  1. Press [Y=]. Enter the given logarithm equation or equations as Y1= and, if needed, Y2=.
  2. Press [GRAPH] to observe the graphs of the curves and use [WINDOW] to find an appropriate view of the graphs, including their point(s) of intersection.
  3. To find the value of x, x, we compute the point of intersection. Press [2ND] then [CALC]. Select “intersect” and press [ENTER] three times. The point of intersection gives the value of x, x, for the point(s) of intersection.

Example 9

Approximating the Solution of a Logarithmic Equation

Solve 4ln( x )+1=2ln( x1 ) 4ln( x )+1=2ln( x1 ) graphically. Round to the nearest thousandth.

Try It #9

Solve 5log( x+2 )=4log( x ) 5log( x+2 )=4log( x ) graphically. Round to the nearest thousandth.

Summarizing Translations of the Logarithmic Function

Now that we have worked with each type of translation for the logarithmic function, we can summarize each in Table 4 to arrive at the general equation for translating exponential functions.

Translations of the Parent Function y= log b ( x ) y= log b ( x )
Translation Form
Shift
  • Horizontally c c units to the left
  • Vertically d d units up
y= log b ( x+c )+d y= log b ( x+c )+d
Stretch and Compress
  • Stretch if | a |>1 | a |>1
  • Compression if | a |<1 | a |<1
y=a log b ( x ) y=a log b ( x )
Reflect about the x-axis y= log b ( x ) y= log b ( x )
Reflect about the y-axis y= log b ( x ) y= log b ( x )
General equation for all translations y=a log b (x+c)+d y=a log b (x+c)+d
Table 4

Translations of Logarithmic Functions

All translations of the parent logarithmic function, y= log b ( x ), y= log b ( x ), have the form

 f(x)=a log b ( x+c )+d  f(x)=a log b ( x+c )+d

where the parent function, y= log b ( x ),b>1, y= log b ( x ),b>1, is

  • shifted vertically up d d units.
  • shifted horizontally to the left c c units.
  • stretched vertically by a factor of | a | | a | if | a |>0. | a |>0.
  • compressed vertically by a factor of | a | | a | if 0<| a |<1. 0<| a |<1.
  • reflected about the x-axis when a<0. a<0.

For f( x )=log( x ), f( x )=log( x ), the graph of the parent function is reflected about the y-axis.

Example 10

Finding the Vertical Asymptote of a Logarithm Graph

What is the vertical asymptote of f(x)=−2 log 3 (x+4)+5? f(x)=−2 log 3 (x+4)+5?

Analysis

The coefficient, the base, and the upward translation do not affect the asymptote. The shift of the curve 4 units to the left shifts the vertical asymptote to x=−4. x=−4.

Try It #10

What is the vertical asymptote of f(x)=3+ln(x1)? f(x)=3+ln(x1)?

Example 11

Finding the Equation from a Graph

Find a possible equation for the common logarithmic function graphed in Figure 15.

Graph of a logarithmic function with a vertical asymptote at x=-2, has been vertically reflected, and passes through the points (-1, 1) and (2, -1).
Figure 15

Analysis

We can verify this answer by comparing the function values in Table 5 with the points on the graph in Figure 15.

x x −1 0 1 2 3
f(x) f(x) 1 0 −0.58496 −1 −1.3219
x x 4 5 6 7 8
f(x) f(x) −1.5850 −1.8074 −2 −2.1699 −2.3219
Table 5

Try It #11

Give the equation of the natural logarithm graphed in Figure 16.

Graph of a logarithmic function with a vertical asymptote at x=-3, has been vertically stretched by 2, and passes through the points (-1, -1).
Figure 16

Q&A

Is it possible to tell the domain and range and describe the end behavior of a function just by looking at the graph?

Yes, if we know the function is a general logarithmic function. For example, look at the graph in Figure 16. The graph approaches x=−3 x=−3 (or thereabouts) more and more closely, so x=−3 x=−3 is, or is very close to, the vertical asymptote. It approaches from the right, so the domain is all points to the right, {x|x>−3}. {x|x>−3}. The range, as with all general logarithmic functions, is all real numbers. And we can see the end behavior because the graph goes down as it goes left and up as it goes right. The end behavior is that as x 3 + ,f(x) x 3 + ,f(x) and as x,f(x). x,f(x).

Media

Access these online resources for additional instruction and practice with graphing logarithms.

6.4 Section Exercises

Verbal

1.

The inverse of every logarithmic function is an exponential function and vice-versa. What does this tell us about the relationship between the coordinates of the points on the graphs of each?

2.

What type(s) of translation(s), if any, affect the range of a logarithmic function?

3.

What type(s) of translation(s), if any, affect the domain of a logarithmic function?

4.

Consider the general logarithmic function f(x)= log b ( x ). f(x)= log b ( x ). Why can’t x x be zero?

5.

Does the graph of a general logarithmic function have a horizontal asymptote? Explain.

Algebraic

For the following exercises, state the domain and range of the function.

6.

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

7.

h(x)=ln( 1 2 x ) h(x)=ln( 1 2 x )

8.

g(x)= log 5 ( 2x+9 )2 g(x)= log 5 ( 2x+9 )2

9.

h(x)=ln( 4x+17 )5 h(x)=ln( 4x+17 )5

10.

f(x)= log 2 ( 123x )3 f(x)= log 2 ( 123x )3

For the following exercises, state the domain and the vertical asymptote of the function.

11.

f(x)= log b (x5) f(x)= log b (x5)

12.

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

13.

f(x)=log(3x+1) f(x)=log(3x+1)

14.

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

15.

g(x)=ln(3x+9)7 g(x)=ln(3x+9)7

For the following exercises, state the domain, vertical asymptote, and end behavior of the function.

16.

f(x)=ln( 2x ) f(x)=ln( 2x )

17.

f(x)=log( x 3 7 ) f(x)=log( x 3 7 )

18.

h(x)=log( 3x4 )+3 h(x)=log( 3x4 )+3

19.

g(x)=ln( 2x+6 )5 g(x)=ln( 2x+6 )5

20.

f(x)= log 3 ( 155x )+6 f(x)= log 3 ( 155x )+6

For the following exercises, state the domain, range, and x- and y-intercepts, if they exist. If they do not exist, write DNE.

21.

h(x)= log 4 ( x1 )+1 h(x)= log 4 ( x1 )+1

22.

f(x)=log( 5x+10 )+3 f(x)=log( 5x+10 )+3

23.

g(x)=ln( x )2 g(x)=ln( x )2

24.

f(x)= log 2 ( x+2 )5 f(x)= log 2 ( x+2 )5

25.

h(x)=3ln( x )9 h(x)=3ln( x )9

Graphical

For the following exercises, match each function in Figure 17 with the letter corresponding to its graph.

Graph of five logarithmic functions.
Figure 17
26.

d(x)=log( x ) d(x)=log( x )

27.

f(x)=ln(x) f(x)=ln(x)

28.

g(x)= log 2 ( x ) g(x)= log 2 ( x )

29.

h(x)= log 5 ( x ) h(x)= log 5 ( x )

30.

j(x)= log 25 ( x ) j(x)= log 25 ( x )

For the following exercises, match each function in Figure 18 with the letter corresponding to its graph.

Graph of three logarithmic functions.
Figure 18
31.

f(x)= log 1 3 ( x ) f(x)= log 1 3 ( x )

32.

g(x)= log 2 ( x ) g(x)= log 2 ( x )

33.

h(x)= log 3 4 ( x ) h(x)= log 3 4 ( x )

For the following exercises, sketch the graphs of each pair of functions on the same axis.

34.

f(x)=log(x) f(x)=log(x) and g(x)= 10 x g(x)= 10 x

35.

f(x)=log(x) f(x)=log(x) and g(x)= log 1 2 (x) g(x)= log 1 2 (x)

36.

f(x)= log 4 (x) f(x)= log 4 (x) and g(x)=ln(x) g(x)=ln(x)

37.

f(x)= e x f(x)= e x and g(x)=ln(x) g(x)=ln(x)

For the following exercises, match each function in Figure 19 with the letter corresponding to its graph.

Graph of three logarithmic functions.
Figure 19
38.

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

39.

g(x)= log 4 ( x+2 ) g(x)= log 4 ( x+2 )

40.

h(x)= log 4 ( x+2 ) h(x)= log 4 ( x+2 )

For the following exercises, sketch the graph of the indicated function.

41.

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

42.

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

43.

f(x)=ln(x) f(x)=ln(x)

44.

g(x)=log( 4x+16 )+4 g(x)=log( 4x+16 )+4

45.

g(x)=log( 63x )+1 g(x)=log( 63x )+1

46.

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

For the following exercises, write a logarithmic equation corresponding to the graph shown.

47.

Use y= log 2 (x) y= log 2 (x) as the parent function.

The graph y=log_2(x) has been reflected over the y-axis and shifted to the right by 1.
48.

Use f(x)= log 3 (x) f(x)= log 3 (x) as the parent function.

The graph y=log_3(x) has been reflected over the x-axis, vertically stretched by 3, and shifted to the left by 4.
49.

Use f(x)= log 4 (x) f(x)= log 4 (x) as the parent function.

The graph y=log_4(x) has been vertically stretched by 3, and shifted to the left by 2.
50.

Use f(x)= log 5 (x) f(x)= log 5 (x) as the parent function.

The graph y=log_3(x) has been reflected over the x-axis and y-axis, vertically stretched by 2, and shifted to the right by 5.

Technology

For the following exercises, use a graphing calculator to find approximate solutions to each equation.

51.

log( x1 )+2=ln( x1 )+2 log( x1 )+2=ln( x1 )+2

52.

log( 2x3 )+2=log( 2x3 )+5 log( 2x3 )+2=log( 2x3 )+5

53.

ln( x2 )=ln( x+1 ) ln( x2 )=ln( x+1 )

54.

2ln( 5x+1 )= 1 2 ln( 5x )+1 2ln( 5x+1 )= 1 2 ln( 5x )+1

55.

1 3 log( 1x )=log( x+1 )+ 1 3 1 3 log( 1x )=log( x+1 )+ 1 3

Extensions

56.

Let b b be any positive real number such that b1. b1. What must log b 1 log b 1 be equal to? Verify the result.

57.

Explore and discuss 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 ). Make a conjecture based on the result.

58.

Prove the conjecture made in the previous exercise.

59.

What is the domain of the function f(x)=ln( x+2 x4 )? f(x)=ln( x+2 x4 )? Discuss the result.

60.

Use properties of exponents to find the x-intercepts of the function f(x)=log( x 2 +4x+4 ) f(x)=log( x 2 +4x+4 ) algebraically. Show the steps for solving, and then verify the result by graphing the function.

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