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Mathematics LibreTexts

4.5 Graphs of Logarithmic Functions

Recall that the exponential function File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image002.gif produces this table of values

x

-3

-2

-1

0

1

2

3

f(x)

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image004.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image006.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image008.gif

1

2

4

8

 

Since the logarithmic function is an inverse of the exponential, File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image010.gif produces the table of values

x

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image004.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image006.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image008.gif

1

2

4

8

g(x)

-3

-2

-1

0

1

2

3

In this second table, notice that

  1. As the input increases, the output increases.
  2. As input increases, the output increases more slowly.
  3. Since the exponential function only outputs positive values, the logarithm can only accept positive values as inputs, so the domain of the log function is File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image012.gif.
  4. Since the exponential function can accept all real numbers as inputs, the logarithm can output any real number, so the range is all real numbers or File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image014.gif.

 

GraphsSketching the graph, notice that as the input approaches zero from the right, the output of the function grows very large in the negative direction, indicating a vertical asymptote at

x = 0.

In symbolic notation we write

as File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image018.gif,  and asFile:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image020.gif

 

 

 

Graphical Features of the Logarithm

  • Graphically, in the function File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image022.gif
  • The graph has a horizontal intercept at (1, 0)
  • The graph has a vertical asymptote at x = 0
  • The graph is increasing and concave down
  • The domain of the function is x > 0, or File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image012.gif
  • The range of the function is all real numbers, or File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image014.gif

 

When sketching a general logarithm with base b, it can be helpful to remember that the graph will pass through the points (1, 0) and (b, 1).

To get a feeling for how the base affects the shape of the graph, examine the graphs below.

 

 
 

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image024.gif

 

 

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image026.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image028.gif

Graphs

Notice that the larger the base, the slower the graph grows.  For example, the common log graph, while it grows without bound, it does so very slowly.  For example, to reach an output of 8, the input must be 100,000,000.

Another important observation made was the domain of the logarithm.  Like the reciprocal and square root functions, the logarithm has a restricted domain which must be considered when finding the domain of a composition involving a log.

 

Example 1

Find the domain of the function File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image032.gif

SOLUTION

The logarithm is only defined with the input is positive, so this function will only be defined when File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image034.gif.  Solving this inequality,

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image036.gif

The domain of this function is File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image038.gif, or in interval notation, File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image040.gif

Try it Now

1. Find the domain of the function File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image042.gif; before solving this as an inequality, consider how the function has been transformed.

Transformations of the Logarithmic Function

Transformations can be applied to a logarithmic function using the basic transformation techniques, but as with exponential functions, several transformations result in interesting relationships.

 

First recall the change of base property tells us that File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image044.gif

From this, we can see that File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image046.gif is a vertical stretch or compression of the graph of the File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image048.gif graph.  This tells us that a vertical stretch or compression is equivalent to a change of base.  For this reason, we typically represent all graphs of logarithmic functions in terms of the common or natural log functions.

Next, consider the effect of a horizontal compression on the graph of a logarithmic function.  Considering File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image050.gif, we can use the sum property to see

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image052.gif

Since log(c) is a constant, the effect of a horizontal compression is the same as the effect of a vertical shift. 

Example 2

Sketch File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image054.gif and File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image056.gif.

SOLUTION

Graphing these,

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image058.gif

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image054.gif

Graphs

Note that, this vertical shift could also be written as a horizontal compression:

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image063.gif.

While a horizontal stretch or compression can be written as a vertical shift, a horizontal reflection is unique and separate from vertical shifting. Finally, we will consider the effect of a horizontal shift on the graph of a logarithm

 

Example 3

Sketch a graph of File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image065.gif.

SOLUTION

This is a horizontal shift to the left by 2 units.  Notice that none of our logarithm rules allow us rewrite this in another form, so the effect of this transformation is unique.  Shifting the graph,

Graphs

Notice that due to the horizontal shift, the vertical asymptote shifted to x = -2, and the domain shifted to File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image069.gif.

Combining these transformations,

Example 4

Sketch a graph of File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image071.gif.

SOLUTION

Factoring the inside as File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image073.gif reveals that this graph is that of the common logarithm, horizontally reflected, vertically stretched by a factor of 5, and shifted to the right by 2 units. 

 

GraphsThe vertical asymptote will be shifted to      x = 2, and the graph will have domain File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image077.gif.  A rough sketch can be created by using the vertical asymptote along with a couple points on the graph, such as

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image079.gif

 

Try it Now

2. Sketch a graph of the function File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image081.gif.

 

Transformations of Logs

Any transformed logarithmic function can be written in the form

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image083.gif,  or File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image085.gif if horizontally reflected,

where

x = b is the vertical asymptote.

 

Example 5

Find an equation for the logarithmic function graphed below.

Graphs

SOLUTION

 

This graph has a vertical asymptote at x = –2 and has been vertically reflected. We do not know yet the vertical shift (equivalent to horizontal stretch) or the vertical stretch (equivalent to a change of base). We know so far that the equation will have form

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image089.gif

It appears the graph passes through the points (–1, 1) and (2, –1). Substituting in (–1, 1),

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image091.gif

Next, substituting in (2, –1),

File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image093.gif

This gives us the equation File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image095.gif

This could also be written as File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image097.gif.

Flashback

3. Write the domain and range of the function graphed in Example 5, and describe its long run behavior.

Important Topics of this Section

  • Graph of the logarithmic function (domain and range)
  • Transformation of logarithmic functions
  • Creating graphs from equations
  • Creating equations from graphs

 

Try it Now Answers

1. Domain:  {x| x > 5} 

2. Graphs

 

Flashback Answers

3.  Domain:  {x|x>-2}, Range: all real numbers;  As File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image101.gifand as File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image103.gif.

Section 4.5 Exercises

 

For each function, find the domain and the vertical asymptote.

1. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image105.gif                                      2. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image107.gif

3. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image109.gif                                        4. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image111.gif

5. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image113.gif                                     6. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image115.gif

7. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image117.gif                                 8. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image119.gif

 

Sketch a graph of each pair of function.

9. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image121.gif                     10. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image123.gif

 

Sketch each transformation.

11. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image125.gif                                       12. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image127.gif

13. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image129.gif                                         14. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image131.gif

15. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image133.gif                                   16. File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image135.gif

 

Find a formula for the transformed logarithm graph shown.

17.File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image137.jpg                           18.File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image139.jpg

 

19.File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image141.jpg                           20.File:/C:\Users\JIMHOR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image143.jpg

           

 

Find a formula for the transformed logarithm graph shown.

 

21.http://www.wamap.org/filter/graph/svgimg.php?sscr=-5%2C5%2C-5%2C5%2C1%2C1%2C1%2C1%2C1%2C200%2C200%2Cfunc%2C3log%28x%2B2%29%2Flog%284%29%2Cnull%2C0%2C0%2C%2C%2Cblue%2C1%2Cnone                           22.http://www.wamap.org/filter/graph/svgimg.php?sscr=-5%2C5%2C-5%2C5%2C1%2C1%2C1%2C1%2C1%2C200%2C200%2Cfunc%2C2log%28x%2B4%29%2Flog%283%29%2Cnull%2C0%2C0%2C%2C%2Cblue%2C1%2Cnone

 

23.http://www.wamap.org/filter/graph/svgimg.php?sscr=-5%2C5%2C-5%2C5%2C1%2C1%2C1%2C1%2C1%2C200%2C200%2Cfunc%2C-2log%28-x%2B5%29%2Flog%285%29%2Cnull%2C0%2C0%2C-6%2C4.999%2Cblue%2C1%2Cnone                           24.http://www.wamap.org/filter/graph/svgimg.php?sscr=-5%2C5%2C-5%2C5%2C1%2C1%2C1%2C1%2C1%2C200%2C200%2Cfunc%2C-4log%28-x%2B3%29%2Flog%286%29%2Cnull%2C0%2C0%2C-6%2C2.999%2Cblue%2C1%2Cnone

Contributors

  • David Lippman (Pierce College)
  • Melonie Rasmussen (Pierce College)