Transformations of Graphs of Logarithmic Functions
Finally, we will transform the graph of logarithmic functions using vertical and horizontal shifts, reflections, and compressions and stretches. Given the graph of a logarithmic function, we will practice defining the equation.
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
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 6.
Figure 6
HORIZONTAL SHIFTS OF THE PARENT FUNCTION
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:
 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 .
EXAMPLE 4
Graphing a Horizontal Shift of the Parent Function
Sketch the horizontal shift alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.
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.
The domain is , the range is , and the vertical asymptote is .
Figure 7
TRY IT #4
Sketch a graph of alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.
Graphing a Vertical Shift of
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 8.
Figure 8
VERTICAL SHIFTS OF THE PARENT FUNCTION
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:
 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 .
EXAMPLE 5
Graphing a Vertical Shift of the Parent Function
Sketch a graph of alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.
Solution
Since the function is , we will notice . Thus .
This means we will shift the function down 2 units.
Consider the three key points from the parent function, , and .
The new coordinates are found by subtracting 2 from the coordinates.
The domain is , the range is , and the vertical asymptote is .
Figure 9
The domain is , the range is , and the vertical asymptote is .
TRY IT #5
Sketch a graph of alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.
Graphing Stretches and Compressions of
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 10.
Figure 10
VERTICAL STRETCHES AND COMPRESSIONS OF THE PARENT FUNCTION
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 intercept .
 has domain .
 has range .
HOW TO
Given a logarithmic function with the form , graph the translation.
 Identify the vertical stretch or compressions:
 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 .
EXAMPLE 6
Graphing a Stretch or Compression of the Parent Function
Sketch a graph of alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.
Solution
Since the function is , we will notice .
This means we will stretch the function by a factor of 2.
Consider the three key points from the parent function, , and .
The new coordinates are found by multiplying the coordinates by 2.
Label the points , (1,0)\), and .
The domain is , the range is , and the vertical asymptote is . See Figure 11.
Figure 11
The domain is , the range is , and the vertical asymptote is .
TRY IT #6
Sketch a graph of 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 . State the domain, range, and asymptote.
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 12. The vertical asymptote will be shifted to . The intercept will be . The domain will be . Two points will help give the shape of the graph: and . We chose as the coordinate of one point to graph because when , the base of the common logarithm.
Figure 12
The domain is , the range is , and the vertical asymptote is .
TRY IT #7
Sketch a graph of the function . State the domain, range, and asymptote.
Graphing Reflections of
When the parent function is multiplied by , the result is a reflection about the axis. When the input is multiplied by , the result is a reflection about the axis. To visualize reflections, we restrict , and observe the general graph of the parent function alongside the reflection about the axis, and the reflection about the axis, .
Figure 13
REFLECTIONS OF THE PARENT FUNCTION
 reflects the parent function about the axis.
 has domain, , range, , and vertical asymptote, , which are unchanged from the parent function.
 reflects the parent function about the axis.
 has domain .
 has range, , and vertical asymptote, , which are unchanged from the parent function.
HOW TO
Given a logarithmic function with the parent function , graph a translation.
Table 3
EXAMPLE 8
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.
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 axis. Thus, will be decreasing as moves from negative infinity to zero, and the right tail of the graph will approach the vertical asymptote .
 The intercept is .
 We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.
Figure 14
The domain is , the range is , and the vertical asymptote is .
TRY IT #8
Graph . State the domain, range, and asymptote.
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.
EXAMPLE 9
Approximating the Solution of a Logarithmic Equation
Solve graphically. Round to the nearest thousandth.
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 â€“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 coordinate 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 #9
Solve 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  
Translation  Form 
Shift 

Stretch and Compress 

Reflect about the axis  
Reflect about the axis  
General equation for all translations 
Table 4
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 axis when .
For , the graph of the parent function is reflected about the axis.
EXAMPLE 10
Finding the Vertical Asymptote of a Logarithm Graph
What is the vertical asymptote of ?
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 #10
What is the vertical asymptote of ?
EXAMPLE 11
Finding the Equation from a Graph
Find a possible equation for the common logarithmic function graphed in Figure 15.
Figure 15
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 ,
Analysis
We can verify this answer by comparing the function values in Table 5 with the points on the graph in Figure 15.
Table 5
TRY IT #11
Give the equation of the natural logarithm graphed in Figure 16.
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 (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 .
Source: Rice University, https://openstax.org/books/collegealgebra/pages/64graphsoflogarithmicfunctions
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