Chapter 9.5: Graphs of Logarithmic Functions
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 in an account that offers an annual interest rate of compounded continuously. We already know that the balance in our account for any year can be found with the equation
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) shows this point on the logarithmic graph.
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 for any real number and constant where
 The domain of is
 The range of is
In the last section we learned that the logarithmic function is the inverse of the exponential function So, as inverse functions:
 The domain of is the range of
 The range of is the domain of
Transformations of the parent function 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 Similarly, applying transformations to the parent function 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 This function is defined for any values of such that the argument, in this case is greater than zero. To find the domain, we set up an inequality and solve for
In interval notation, the domain of is
How To
Given a logarithmic function, identify the domain.
 Set up an inequality showing the argument greater than zero.
 Solve for
 Write the domain in interval notation.
Identifying the Domain of a Logarithmic Shift
What is the domain of
Show Solution
The logarithmic function is defined only when the input is positive, so this function is defined when Solving this inequality,
The domain of is
Try It
What is the domain of
Show Solution
Identifying the Domain of a Logarithmic Shift and Reflection
What is the domain of
Show Solution
The logarithmic function is defined only when the input is positive, so this function is defined when Solving this inequality,
The domain of is
Try It
What is the domain of
Show Solution
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 along with all its transformations: shifts, stretches, compressions, and reflections.
We begin with the parent function Because every logarithmic function of this form is the inverse of an exponential function with the form their graphs will be reflections of each other across the line To illustrate this, we can observe the relationship between the input and output values of and its equivalent in (Figure).
Using the inputs and outputs from (Figure), we can build another table to observe the relationship between points on the graphs of the inverse functions and See (Figure).
As we’d expect, the x– and ycoordinates are reversed for the inverse functions. (Figure) shows the graph of and
Observe the following from the graph:
 has a yintercept at and has an x– intercept at
 The domain of is the same as the range of
 The range of is the same as the domain of
Characteristics of the Graph of the Parent Function, f(x) = log_{b}(x)
For any real number and constant we can see the following characteristics in the graph of
 onetoone function
 vertical asymptote:
 domain:
 range:
 xintercept: and key point
 yintercept: none
 increasing if
 decreasing if
See (Figure).
(Figure) shows how changing the base in can affect the graphs. Observe that the graphs compress vertically as the value of the base increases. (Note: recall that the function has base
How To
Given a logarithmic function with the form graph the function.
 Draw and label the vertical asymptote,
 Plot the xintercept,
 Plot the key point
 Draw a smooth curve through the points.
 State the domain, the range, and the vertical asymptote,
Graphing a Logarithmic Function with the Form f(x) = log_{b}(x).
Graph State the domain, range, and asymptote.
Show Solution
Before graphing, identify the behavior and key points for the graph.
 Since is greater than one, we know the function is increasing. The left tail of the graph will approach the vertical asymptote and the right tail will increase slowly without bound.
 The xintercept is
 The key point is on the graph.
 We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points (see (Figure)).
The domain is the range is and the vertical asymptote is
Try It
Graph State the domain, range, and asymptote.
Show Solution
The domain is the range is and the vertical asymptote is
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 without loss of shape.
Graphing a Horizontal Shift of f(x) = log_{b}(x)
When a constant is added to the input of the parent function the result is a horizontal shift units in the opposite direction of the sign on To visualize horizontal shifts, we can observe the general graph of the parent function and for alongside the shift left, and the shift right, See (Figure).
Horizontal Shifts of the Parent Function y = log_{b}(x)
For any constant the function
 shifts the parent function left units if
 shifts the parent function right units if
 has the vertical asymptote
 has domain
 has range
How To
Given a logarithmic function with the form graph the translation.
 Identify the horizontal shift:
 If shift the graph of left units.
 If shift the graph of right units.
 Draw the vertical asymptote
 Identify three key points from the parent function. Find new coordinates for the shifted functions by subtracting from the coordinate.
 Label the three points.
 The Domain is the range is and the vertical asymptote is
Graphing a Horizontal Shift of the Parent Function y = log_{b}(x)
Sketch the horizontal shift alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.
Show Solution
Since the function is we notice
Thus so This means we will shift the function right 2 units.
The vertical asymptote is or
Consider the three key points from the parent function, and
The new coordinates are found by adding 2 to the coordinates.
Label the points and
The domain is the range is and the vertical asymptote is
Try It
Sketch a graph of alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.
Show Solution
The domain is the range and the asymptote
Graphing a Vertical Shift of y = log_{b}(x)
When a constant is added to the parent function the result is a vertical shift units in the direction of the sign on To visualize vertical shifts, we can observe the general graph of the parent function alongside the shift up, and the shift down, See (Figure).
Vertical Shifts of the Parent Function y = log_{b}(x)
For any constant the function
 shifts the parent function up units if
 shifts the parent function down units if
 has the vertical asymptote
 has domain
 has range
How To
Given a logarithmic function with the form graph the translation.
 Identify the vertical shift:
 If shift the graph of up units.
 If shift the graph of down units.
 Draw the vertical asymptote
 Identify three key points from the parent function. Find new coordinates for the shifted functions by adding to the coordinate.
 Label the three points.
 The domain is the range is and the vertical asymptote is
Graphing a Vertical Shift of the Parent Function y = log_{b}(x)
Sketch a graph of alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.
Show Solution
Since the function is we will notice Thus
This means we will shift the function down 2 units.
The vertical asymptote is
Consider the three key points from the parent function, and
The new coordinates are found by subtracting 2 from the y coordinates.
Label the points and
The domain is the range is and the vertical asymptote is
The domain is the range is and the vertical asymptote is
Try It
Sketch a graph of alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.
Show Solution
The domain is the range is and the vertical asymptote is
Graphing Stretches and Compressions of y = log_{b}(x)
When the parent function is multiplied by a constant the result is a vertical stretch or compression of the original graph. To visualize stretches and compressions, we set and observe the general graph of the parent function alongside the vertical stretch, and the vertical compression, See (Figure).
Vertical Stretches and Compressions of the Parent Function y = log_{b}(x)
For any constant the function
 stretches the parent function vertically by a factor of if
 compresses the parent function vertically by a factor of if
 has the vertical asymptote
 has the xintercept
 has domain
 has range
Given a logarithmic function with the form graph the translation.
 Identify the vertical stretch or compressions:
 If the graph of is stretched by a factor of units.
 If the graph of is compressed by a factor of units.
 Draw the vertical asymptote
 Identify three key points from the parent function. Find new coordinates for the shifted functions by multiplying the coordinates by
 Label the three points.
 The domain is the range is and the vertical asymptote is
Graphing a Stretch or Compression of the Parent Function y = log_{b}(x)
Sketch a graph of alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.
Show Solution
Since the function is we will notice
This means we will stretch the function by a factor of 2.
The vertical asymptote is
Consider the three key points from the parent function, and
The new coordinates are found by multiplying the coordinates by 2.
Label the points and
The domain is the range is and the vertical asymptote is See (Figure).
The domain is the range is and the vertical asymptote is
Try It
Sketch a graph of alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.
Show Solution
The domain is the range is and the vertical asymptote is
Combining a Shift and a Stretch
Sketch a graph of State the domain, range, and asymptote.
Show Solution
Remember: what happens inside parentheses happens first. First, we move the graph left 2 units, then stretch the function vertically by a factor of 5, as in (Figure). The vertical asymptote will be shifted to The xintercept will be The domain will be Two points will help give the shape of the graph: and We chose as the xcoordinate of one point to graph because when the base of the common logarithm.
The domain is the range is and the vertical asymptote is
Try It
Sketch a graph of the function State the domain, range, and asymptote.
Show Solution
The domain is the range is and the vertical asymptote is
Graphing Reflections of f(x) = log_{b}(x)
When the parent function is multiplied by the result is a reflection about the xaxis. When the input is multiplied by the result is a reflection about the yaxis. To visualize reflections, we restrict and observe the general graph of the parent function alongside the reflection about the xaxis, and the reflection about the yaxis,
Reflections of the Parent Function y = log_{b}(x)
The function
 reflects the parent function about the xaxis.
 has domain, range, and vertical asymptote, which are unchanged from the parent function.
The function
 reflects the parent function about the yaxis.
 has domain
 has range, and vertical asymptote, which are unchanged from the parent function.
Given a logarithmic function with the parent function graph a translation.










Graphing a Reflection of a Logarithmic Function
Sketch a graph of alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.
Show Solution
Before graphing identify the behavior and key points for the graph.
 Since is greater than one, we know that the parent function is increasing. Since the input value is multiplied by is a reflection of the parent graph about the yaxis. Thus, will be decreasing as moves from negative infinity to zero, and the right tail of the graph will approach the vertical asymptote
 The xintercept is
 We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.
The domain is the range is and the vertical asymptote is
Try It
Graph State the domain, range, and asymptote.
Show Solution
The domain is the range is and the vertical asymptote is
How To
Given a logarithmic equation, use a graphing calculator to approximate solutions.
 Press [Y=]. Enter the given logarithm equation or equations as Y_{1}= and, if needed, Y_{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.
 To find the value of 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 for the point(s) of intersection.
Approximating the Solution of a Logarithmic Equation
Solve graphically. Round to the nearest thousandth.
Show Solution
Press [Y=] and enter next to Y_{1}=. Then enter next to Y_{2}=. For a window, use the values 0 to 5 for and –10 to 10 for Press [GRAPH]. The graphs should intersect somewhere a little to right of
For a better approximation, press [2ND] then [CALC]. Select [5: intersect] and press [ENTER] three times. The xcoordinate of the point of intersection is displayed as 1.3385297. (Your answer may be different if you use a different window or use a different value for Guess?) So, to the nearest thousandth,
Try It
Solve graphically. Round to the nearest thousandth.
Show Solution
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 (Figure) to arrive at the general equation for translating exponential functions.
Translations of the Parent Function  

Translation  Form 
Shift


Stretch and Compress


Reflect about the xaxis  
Reflect about the yaxis  
General equation for all translations 
Translations of Logarithmic Functions
All translations of the parent logarithmic function, have the form
where the parent function, is
 shifted vertically up units.
 shifted horizontally to the left units.
 stretched vertically by a factor of if
 compressed vertically by a factor of if
 reflected about the xaxis when
For the graph of the parent function is reflected about the yaxis.
Finding the Vertical Asymptote of a Logarithm Graph
What is the vertical asymptote of
Show Solution
The vertical asymptote is at
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
Try It
What is the vertical asymptote of
Show Solution
Finding the Equation from a Graph
Find a possible equation for the common logarithmic function graphed in (Figure).
Show Solution
This graph has a vertical asymptote at and has been vertically reflected. We do not know yet the vertical shift or the vertical stretch. We know so far that the equation will have form:
It appears the graph passes through the points and Substituting
Next, substituting in ,
This gives us the equation
Try It
Give the equation of the natural logarithm graphed in (Figure).
Show Solution
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). The graph approaches (or thereabouts) more and more closely, so is, or is very close to, the vertical asymptote. It approaches from the right, so the domain is all points to the right, 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 and as
Access these online resources for additional instruction and practice with graphing logarithms.
Key Equations
General Form for the Translation of the Parent Logarithmic Function 
Key Concepts
 To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero, and solve for See (Figure) and (Figure)
 The graph of the parent function has an xintercept at domain range vertical asymptote and
 if the function is increasing.
 if the function is decreasing.
See (Figure).
 The equation shifts the parent function horizontally
 left units if
 right units if
See (Figure).
 The equation shifts the parent function vertically
 up units if
 down units if
See (Figure).
 For any constant the equation
 stretches the parent function vertically by a factor of if
 compresses the parent function vertically by a factor of if
 When the parent function is multiplied by the result is a reflection about the xaxis. When the input is multiplied by the result is a reflection about the yaxis.
 The equation represents a reflection of the parent function about the xaxis.
 The equation represents a reflection of the parent function about the yaxis.
See (Figure).
 A graphing calculator may be used to approximate solutions to some logarithmic equations See (Figure).
 All translations of the logarithmic function can be summarized by the general equation See (Figure).
 Given an equation with the general form we can identify the vertical asymptote for the transformation. See (Figure).
 Using the general equation we can write the equation of a logarithmic function given its graph. See (Figure).
Section Exercises
Verbal
 The inverse of every logarithmic function is an exponential function and viceversa. What does this tell us about the relationship between the coordinates of the points on the graphs of each?
Show Solution
Since the functions are inverses, their graphs are mirror images about the line So for every point on the graph of a logarithmic function, there is a corresponding point on the graph of its inverse exponential function.
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?
Show Solution
Shifting the function right or left and reflecting the function about the yaxis will affect its domain.
4. Consider the general logarithmic function Why can’t be zero?
5. Does the graph of a general logarithmic function have a horizontal asymptote? Explain.
Show Solution
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.
Algebraic
For the following exercises, state the domain and range of the function.
7.
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Domain: Range:
8.
9.
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Domain: Range:
10.
For the following exercises, state the domain and the vertical asymptote of the function.
11.
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Domain: Vertical asymptote:
12.
13.
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Domain: Vertical asymptote:
14.
15.
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Domain: Vertical asymptote:
For the following exercises, state the domain, vertical asymptote, and end behavior of the function.
16.
17.
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Domain: ;
18.
19.
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Domain: ; Vertical asymptote: ;
20.
For the following exercises, state the domain, range, and x– and yintercepts, if they exist. If they do not exist, write DNE.
21.
Show Solution
Domain: Range: Vertical asymptote: xintercept: yintercept: DNE
22.
23.
Show Solution
Domain: Range: Vertical asymptote: xintercept: yintercept: DNE
24.
Show Solution
Domain: Range: Vertical asymptote: xintercept: yintercept: DNE
Graphical
For the following exercises, match each function in (Figure) with the letter corresponding to its graph.
26.
27.
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B
28.
29.
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C
30.
For the following exercises, match each function in (Figure) with the letter corresponding to its graph.
31.
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B
32.
33.
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C
For the following exercises, sketch the graphs of each pair of functions on the same axis.
34. and
35. and
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36. and
37. and
Show Solution
For the following exercises, match each function in (Figure) with the letter corresponding to its graph.
38.
39.
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C
40.
For the following exercises, sketch the graph of the indicated function.
41.
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42.
43.
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44.
45.
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46.
For the following exercises, write a logarithmic equation corresponding to the graph shown.
47. Use as the parent function.
Show Solution
48. Use as the parent function.
49. Use as the parent function.
Show Solution
50. Use as the parent function.
Technology
For the following exercises, use a graphing calculator to find approximate solutions to each equation.
51.
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52.
53.
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54.
55.
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Extensions
56. Let be any positive real number such that What must be equal to? Verify the result.
57. Explore and discuss the graphs of and Make a conjecture based on the result.
Show Solution
The graphs of and appear to be the same; Conjecture: for any positive base
58. Prove the conjecture made in the previous exercise.
59. What is the domain of the function Discuss the result.
Show Solution
Recall that the argument of a logarithmic function must be positive, so we determine where . From the graph of the function note that the graph lies above the xaxis on the interval and again to the right of the vertical asymptote, that is Therefore, the domain is
60. Use properties of exponents to find the xintercepts of the function algebraically. Show the steps for solving, and then verify the result by graphing the function.