Stretch, Compress, or Reflect a Logarithmic Function
Learning Objectives
 Graph stretches and compressions of logarithmic functions
 Graph reflections of logarithmic functions
Graphing Stretches and Compressions of [latex]y=\text{log}_{b}\left(x\right)[/latex]
When the parent function [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex] is multiplied by a constant a > 0, the result is a vertical stretch or compression of the original graph. To visualize stretches and compressions, we set a > 1 and observe the general graph of the parent function [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex] alongside the vertical stretch, [latex]g\left(x\right)=a{\mathrm{log}}_{b}\left(x\right)[/latex] and the vertical compression, [latex]h\left(x\right)=\frac{1}{a}{\mathrm{log}}_{b}\left(x\right)[/latex].try it
In the graph below, you can explore how to produce vertical stretches and compressions of a logarithmic function. The function [latex]g(x) = a\log_{b}{x}[/latex] represents a vertical stretch, and the function [latex]f(x) = \frac{1}{a}\log_{b}{x}[/latex] represents a vertical compression. As you investigate, consider the following questions; Both the vertical stretch and compression produce graphs that are increasing. Which transformation produces a function that increases faster?
 One of the key points that is commonly defined for transformations of a logarithmic function comes from finding the input that gives an output of [latex]y = 1[/latex]. This point can help you determine whether a graph is the result of a vertical compression or stretch. Explain why.
A General Note: Vertical Stretches and Compressions of the Parent Function [latex]y=\text{log}_{b}\left(x\right)[/latex]
For any constant a > 1, the function [latex]f\left(x\right)=a{\mathrm{log}}_{b}\left(x\right)[/latex] stretches the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] vertically by a factor of a if a > 1.
 compresses the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] vertically by a factor of a if 0 < a < 1.
 has the vertical asymptote x = 0.
 has the xintercept [latex]\left(1,0\right)[/latex].
 has domain [latex]\left(0,\infty \right)[/latex].
 has range [latex]\left(\infty ,\infty \right)[/latex].
How To: Given a logarithmic function with the form [latex]f\left(x\right)=a{\mathrm{log}}_{b}\left(x\right)[/latex], [latex]a>0[/latex], graph the translation.
 Identify the vertical stretch or compressions:
 If [latex]a>1[/latex], the graph of [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex] is stretched by a factor of a units.
 If [latex]a<1[/latex], the graph of [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex] is compressed by a factor of a units.
 Draw the vertical asymptote x = 0.
 Identify three key points from the parent function. Find new coordinates for the shifted functions by multiplying the y coordinates by a.
 Label the three points.
 The domain is [latex]\left(0,\infty \right)[/latex], the range is [latex]\left(\infty ,\infty \right)[/latex], and the vertical asymptote is x = 0.
Example: Graphing a Stretch or Compression of the Parent Function [latex]y=\text{log}_{b}\left(x\right)[/latex]
Sketch a graph of [latex]f\left(x\right)=2{\mathrm{log}}_{4}\left(x\right)[/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.Answer: Since the function is [latex]f\left(x\right)=2{\mathrm{log}}_{4}\left(x\right)[/latex], we will notice a = 2. This means we will stretch the function [latex]f\left(x\right)={\mathrm{log}}_{4}\left(x\right)[/latex] by a factor of 2. The vertical asymptote is x = 0. Consider the three key points from the parent function, [latex]\left(\frac{1}{4},1\right)[/latex], [latex]\left(1,0\right)[/latex], and [latex]\left(4,1\right)[/latex]. The new coordinates are found by multiplying the y coordinates by 2. Label the points [latex]\left(\frac{1}{4},2\right)[/latex], [latex]\left(1,0\right)[/latex], and [latex]\left(4,\text{2}\right)[/latex].
Try It
Sketch a graph of [latex]f\left(x\right)=\frac{1}{2}{\mathrm{log}}_{4}\left(x\right)[/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.Answer: The domain is [latex]\left(0,\infty \right)\\[/latex], the range is [latex]\left(\infty ,\infty \right)\\[/latex], and the vertical asymptote is x = 0.
Example: Combining a Shift and a Stretch
Sketch a graph of [latex]f\left(x\right)=5\mathrm{log}\left(x+2\right)[/latex]. State the domain, range, and asymptote.Answer: 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. The vertical asymptote will be shifted to x = –2. The xintercept will be [latex]\left(1,0\right)[/latex]. The domain will be [latex]\left(2,\infty \right)[/latex]. Two points will help give the shape of the graph: [latex]\left(1,0\right)[/latex] and [latex]\left(8,5\right)[/latex]. We chose x = 8 as the xcoordinate of one point to graph because when x = 8, x + 2 = 10, the base of the common logarithm.
Try It
Sketch a graph of the function [latex]f\left(x\right)=3\mathrm{log}\left(x  2\right)+1[/latex]. State the domain, range, and asymptote.Answer: The domain is [latex]\left(2,\infty \right)[/latex], the range is [latex]\left(\infty ,\infty \right)[/latex], and the vertical asymptote is x = 2.
Graphing Reflections of [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex]
When the parent function [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex] is multiplied by –1, the result is a reflection about the xaxis. When the input is multiplied by –1, the result is a reflection about the yaxis. To visualize reflections, we restrict b > 1, and observe the general graph of the parent function [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex] alongside the reflection about the xaxis, [latex]g\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex] and the reflection about the yaxis, [latex]h\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex].Try it
The functions [latex]f(x) = \log_{b}{x},g(x) = \log_{b}{x}, h(x) = \log_{b}{x} [/latex] are graphed below. The point of interest for the function [latex]f(x)[/latex] has also been added. Consider the following questions: Which graph, [latex]g(x) = \log_{b}{x}, \text{ or }h(x) = \log_{b}{x}[/latex] represents a vertical reflection? Which one represents a horizontal reflection?
 The point [latex](b,1)[/latex] has been added as a point of interest for the function [latex]f(x)[/latex]. Using the variable b as your x value, add the corresponding points of interest for [latex]g(x),\text{ and }h(x)[/latex].
 Does the vertical asymptote change when you reflect the graph of [latex]f(x)[/latex] either vertically or horizontally?
A General Note: Reflections of the Parent Function [latex]y=\text{log}_{b}\left(x\right)[/latex]
The function [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex] reflects the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] about the xaxis.
 has domain, [latex]\left(0,\infty \right)[/latex], range, [latex]\left(\infty ,\infty \right)[/latex], and vertical asymptote, x = 0, which are unchanged from the parent function.
 reflects the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] about the yaxis.
 has domain [latex]\left(\infty ,0\right)[/latex].
 has range, [latex]\left(\infty ,\infty \right)[/latex], and vertical asymptote, x = 0, which are unchanged from the parent function.
How To: Given a logarithmic function with the parent function [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex], graph a translation.
[latex]\text{If }f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex]  [latex]\text{If }f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex] 

1. Draw the vertical asymptote, x = 0.  1. Draw the vertical asymptote, x = 0. 
2. Plot the xintercept, [latex]\left(1,0\right)[/latex].  2. Plot the xintercept, [latex]\left(1,0\right)[/latex]. 
3. Reflect the graph of the parent function [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex] about the xaxis.  3. Reflect the graph of the parent function [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex] about the yaxis. 
4. Draw a smooth curve through the points.  4. Draw a smooth curve through the points. 
5. State the domain, [latex]\left(0,\infty \right)[/latex], the range, [latex]\left(\infty ,\infty \right)[/latex], and the vertical asymptote x = 0.  5. State the domain, [latex]\left(\infty ,0\right)[/latex], the range, [latex]\left(\infty ,\infty \right)[/latex], and the vertical asymptote x = 0. 
Example: Graphing a Reflection of a Logarithmic Function
Sketch a graph of [latex]f\left(x\right)=\mathrm{log}\left(x\right)[/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.Answer: Before graphing [latex]f\left(x\right)=\mathrm{log}\left(x\right)[/latex], identify the behavior and key points for the graph.
 Since b = 10 is greater than one, we know that the parent function is increasing. Since the input value is multiplied by –1, f is a reflection of the parent graph about the yaxis. Thus, [latex]f\left(x\right)=\mathrm{log}\left(x\right)[/latex] will be decreasing as x moves from negative infinity to zero, and the right tail of the graph will approach the vertical asymptote x = 0.
 The xintercept is [latex]\left(1,0\right)[/latex].
 We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.
Try It
Graph [latex]f\left(x\right)=\mathrm{log}\left(x\right)[/latex]. State the domain, range, and asymptote.Answer: The domain is [latex]\left(\infty ,0\right)[/latex], the range is [latex]\left(\infty ,\infty \right)[/latex], and the vertical asymptote is x = 0.
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 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, for the point(s) of intersection.
Example: Approximating the Solution of a Logarithmic Equation
Solve [latex]4\mathrm{ln}\left(x\right)+1=2\mathrm{ln}\left(x  1\right)[/latex] graphically. Round to the nearest thousandth.Answer: Press [Y=] and enter [latex]4\mathrm{ln}\left(x\right)+1[/latex] next to Y_{1}=. Then enter [latex]2\mathrm{ln}\left(x  1\right)[/latex] next to Y_{2}=. For a window, use the values 0 to 5 for x and –10 to 10 for y. Press [GRAPH]. The graphs should intersect somewhere a little to right of x = 1. 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, [latex]x\approx 1.339[/latex].
Try It
Solve [latex]5\mathrm{log}\left(x+2\right)=4\mathrm{log}\left(x\right)[/latex] graphically. Round to the nearest thousandth.Answer: [latex]x\approx 3.049[/latex]
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 the table below to arrive at the general equation for translating exponential functions.Translations of the Parent Function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex]  

Translation  Form 
Shift

[latex]y={\mathrm{log}}_{b}\left(x+c\right)+d[/latex] 
Stretch and Compress

[latex]y=a{\mathrm{log}}_{b}\left(x\right)[/latex] 
Reflect about the xaxis  [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] 
Reflect about the yaxis  [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] 
General equation for all translations  [latex]y=a{\mathrm{log}}_{b}\left(x+c\right)+d[/latex] 
A General Note: Translations of Logarithmic Functions
All translations of the parent logarithmic function, [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex], have the form [latexdisplay] f\left(x\right)=a{\mathrm{log}}_{b}\left(x+c\right)+d[/latexdisplay] where the parent function, [latex]y={\mathrm{log}}_{b}\left(x\right),b>1[/latex], is shifted vertically up d units.
 shifted horizontally to the left c units.
 stretched vertically by a factor of a if a > 0.
 compressed vertically by a factor of a if 0 < a < 1.
 reflected about the xaxis when a < 0.
Example: Finding the Vertical Asymptote of a Logarithm Graph
What is the vertical asymptote of [latex]f\left(x\right)=2{\mathrm{log}}_{3}\left(x+4\right)+5[/latex]?Answer: The vertical asymptote is at x = –4.
Analysis of the Solution
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.Try It
What is the vertical asymptote of [latex]f\left(x\right)=3+\mathrm{ln}\left(x  1\right)[/latex]?Answer: [latex]x=1[/latex]
Example: Finding the Equation from a Graph
Find a possible equation for the common logarithmic function graphed below.Answer: This graph has a vertical asymptote at x = –2 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:
[latex]f\left(x\right)=a\mathrm{log}\left(x+2\right)+k[/latex]
It appears the graph passes through the points [latex]\left(1,1\right)[/latex] and [latex]\left(2,1\right)[/latex]. Substituting [latex]\left(1,1\right)[/latex],[latex]\begin{cases}1=a\mathrm{log}\left(1+2\right)+k\hfill & \text{Substitute }\left(1,1\right).\hfill \\ 1=a\mathrm{log}\left(1\right)+k\hfill & \text{Arithmetic}.\hfill \\ 1=k\hfill & \text{log(1)}=0.\hfill \end{cases}[/latex]
Next, substituting in [latex]\left(2,1\right)[/latex],[latex]\begin{cases}1=a\mathrm{log}\left(2+2\right)+1\hfill & \hfill & \text{Plug in }\left(2,1\right).\hfill \\ 2=a\mathrm{log}\left(4\right)\hfill & \hfill & \text{Arithmetic}.\hfill \\ \text{ }a=\frac{2}{\mathrm{log}\left(4\right)}\hfill & \hfill & \text{Solve for }a.\hfill \end{cases}[/latex]
This gives us the equation [latex]f\left(x\right)=\frac{2}{\mathrm{log}\left(4\right)}\mathrm{log}\left(x+2\right)+1[/latex].Analysis of the Solution
We can verify this answer by comparing the function values in the table below with the points on the graph in this example.x  −1  0  1  2  3 
f(x)  1  0  −0.58496  −1  −1.3219 
x  4  5  6  7  8 
f(x)  −1.5850  −1.8074  −2  −2.1699  −2.3219 
Try It
Give the equation of the natural logarithm graphed below.Answer: [latex]f\left(x\right)=2\mathrm{ln}\left(x+3\right)1[/latex]
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 the last Try It. The graph approaches x = –3 (or thereabouts) more and more closely, so 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, [latex]\left\{xx>3\right\}[/latex]. 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 [latex]x\to {3}^{+},f\left(x\right)\to \infty [/latex] and as [latex]x\to \infty ,f\left(x\right)\to \infty [/latex].Licenses & Attributions
CC licensed content, Original
 Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
 Reflections of Logarithmic Functions Interactive. Authored by: Lumen Learning. Located at: https://www.desmos.com/calculator/5qmaorrlst. License: Public Domain: No Known Copyright.
 Vertical Stretch and Compression of a Logarithmic Function Interactive. Authored by: Lumen Learning. Located at: https://www.desmos.com/calculator/arlodqq7be. License: Public Domain: No Known Copyright.
CC licensed content, Shared previously
 College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. Located at: https://openstax.org/books/collegealgebra/pages/1introductiontoprerequisites. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].