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=logb(x) along with all its transformations: shifts, stretches, compressions, and reflections.
We begin with the parent function y=logb(x). Because every logarithmic function of this form is the inverse of an exponential function with the form y=bx, their graphs will be reflections of each other across the line y=x. To illustrate this, we can observe the relationship between the input and output values of y=2x and its equivalent x=log2(y) in the table below.
x
–3
–2
–1
0
1
2
3
2x=y
81
41
21
1
2
4
8
log2(y)=x
–3
–2
–1
0
1
2
3
Using the inputs and outputs from the table above, we can build another table to observe the relationship between points on the graphs of the inverse functions f(x)=2x and g(x)=log2(x).
f(x)=2x
(−3,81)
(−2,41)
(−1,21)
(0,1)
(1,2)
(2,4)
(3,8)
g(x)=log2(x)
(81,−3)
(41,−2)
(21,−1)
(1,0)
(2,1)
(4,2)
(8,3)
As we’d expect, the x- and y-coordinates are reversed for the inverse functions. The figure below shows the graph of f and g.
Figure 2. Notice that the graphs of f(x)=2x and g(x)=log2(x) are reflections about the line y = x.
Observe the following from the graph:
f(x)=2x has a y-intercept at (0,1) and g(x)=log2(x) has an x-intercept at (1,0).
The domain of f(x)=2x, (−∞,∞), is the same as the range of g(x)=log2(x).
The range of f(x)=2x, (0,∞), is the same as the domain of g(x)=log2(x).
A General Note: Characteristics of the Graph of the Parent Function, f(x) = logb(x)
For any real number x and constant b > 0, b=1, we can see the following characteristics in the graph of f(x)=logb(x):
one-to-one function
vertical asymptote: x = 0
domain: (0,∞)
range: (−∞,∞)
x-intercept: (1,0) and key point (b,1)
y-intercept: none
increasing if b>1
decreasing if 0 < b < 1
Figure 3
Figure 3 shows how changing the base b in f(x)=logb(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) has base e≈2.718.)
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)=logb(x), graph the function.
Draw and label the vertical asymptote, x = 0.
Plot the x-intercept, (1,0).
Plot the key point (b,1).
Draw a smooth curve through the points.
State the domain, (0,∞), the range, (−∞,∞), and the vertical asymptote, x = 0.
Example 3: Graphing a Logarithmic Function with the Form f(x)=logb(x).
Graph f(x)=log5(x). State the domain, range, and asymptote.
Solution
Before graphing, identify the behavior and key points for the graph.
Since b = 5 is greater than one, we know the function is increasing. The left tail of the graph will approach the vertical asymptote x = 0, and the right tail will increase slowly without bound.
The x-intercept is (1,0).
The key point (5,1) is on the graph.
We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.
Figure 5. The domain is (0,∞), the range is (−∞,∞), and the vertical asymptote is x = 0.
Try It 3
Graph f(x)=log51(x). State the domain, range, and asymptote.
Solution
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Precalculus.Provided by: OpenStaxAuthored by: Jay Abramson, et al..Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions.License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175..