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Graphs of Logarithmic Functions

Author: Sophia

what's covered

In this lesson, you will learn properties of the graph of a logarithmic function and determine its domain and range. Specifically, this lesson will cover:

1. The Graph of a Logarithmic Function Where b > 1
2. The Graph of a Logarithmic Function Where 0 < b < 1
3. Comparing Graphs of Logarithmic Functions

1. The Graph of a Logarithmic Function Where b > 1

Consider the function f open parentheses x close parentheses equals 2 to the power of x. A table of values as well as its graph are shown below.


-3
-2
-1
0
1
2
3
4

A graph with an x-axis ranging from −8 to 8 and a y-axis ranging from −1 to 11. The graph has a curve that rises along the negative x-axis, opens upward, and extends upward into the first quadrant by passing through the marked points at (−3, 1 over 8), (−2, 1 over 4), (−1, 1 over 2), (0, 1), (1, 2), (2, 4), and (3, 8).

As from above, and as
has a horizontal asymptote of
Note that the domain is and the range is

We know that the inverse of f open parentheses x close parentheses

g open parentheses x close parentheses equals log subscript 2 x.

Its table of values and graph are shown below.


	-3
	-2
	-1
	0
	1
	2
	3
	4

A graph with an x-axis ranging from −2 to 12 and a y-axis ranging from −6 to 5. The graph has a curve that rises along the negative y-axis, opens downward, and extends into the first quadrant, passing through the marked points at (1 over 8, −3), (1 over 4, −2), (1 over 2, −1), (1, 0), (2, 1), (4, 2), and (8, 3).

As from the right, , and as
has a vertical asymptote of
Since this is the inverse of an exponential function, the domain is and the range is

When

the graph of y equals b to the power of x

has a shape similar to the graph of f open parentheses x close parentheses equals 2 to the power of x.

This means that the graph of y equals log subscript b x comma

when

will look similar to the graph of g open parentheses x close parentheses equals log subscript 2 x.

This includes y equals ln x comma

since e is approximately 2.71828, which is greater than 1, and y equals log x comma

which is base 10.

big idea

When

and

the graph of f open parentheses x close parentheses equals b to the power of x

passes through the point open parentheses 0 comma space 1 close parentheses.

Then, the graph of its inverse, g open parentheses x close parentheses equals log subscript b x

, passes through the point open parentheses 1 comma space 0 close parentheses.

EXAMPLE

Sketch the graph of f open parentheses x close parentheses equals ln x.

Recall that the output of a logarithmic function is the exponent of the input, when the base of the input matches the base of the logarithm. Consider this table of values.

				1
Approximate	0.0498	0.1353	0.3679	1	2.7183	7.3891	20.0855
	-3	-2	-1	0	1	2	3

In the figure, we have the graph of f open parentheses x close parentheses equals ln x.

Note the vertical asymptote x equals 0.

A graph with an x-axis ranging from −1 to 12 and a y-axis ranging from −4 to 3. The graph has a curve that rises along the negative y-axis, opens downward, and extends into the first quadrant by passing through the marked points labeled (e to the power −3, −3), (e to the power −2, −2), (e to the power −1, −1), (1, 0), (e to the power 1, 1), and (e to the power 2, 2).

2. The Graph of a Logarithmic Function Where 0 < b < 1

Consider the function f open parentheses x close parentheses equals open parentheses 1 third close parentheses to the power of x. A table of values as well as its graph are shown below.


-3	27
-2	9
-1	3
0	1
1
2
3

A graph with an x-axis ranging from −4 to 4 and a y-axis ranging from 0 to 25 in increments of 5. The graph has a curve that descends from the second quadrant, opens upward, and extends into the first quadrant along the positive x-axis by passing through the marked points at (−3, 27), (−2, 9), (−1, 3), (0, 1), (1, 1 over 3), (2, 1 over 9), and (3, 1 over 27).

As , and as from above.
has a horizontal asymptote of
Note that the domain is and the range is

We know that the inverse of f open parentheses x close parentheses

g open parentheses x close parentheses equals log subscript 1 divided by 3 end subscript x.

Its table of values and graph are shown below.


	3
	2
	1
1	0
3	-1
9	-2
27	-3

A graph with an x-axis ranging from −5 to 50 in increments of 5 and a y-axis ranging from −4 to 4. The graph has a curve that descends along the positive y-axis, opens upward, and extends into the fourth quadrant, and passes through the marked points at (1 over 27, 3), (1 over 9, 2), (1 over 3, 1), (1, 0), (3, −1), (9, −2), and (27, −3).

As from the right, , and as
has a vertical asymptote of
Since this is the inverse of an exponential function, the domain is and the range is

When

the graph of y equals b to the power of x

has a shape similar to the graph of f open parentheses x close parentheses equals open parentheses 1 third close parentheses to the power of x.

This means that the graph of y equals log subscript b x comma

when

will look similar to the graph of g open parentheses x close parentheses equals log subscript 1 divided by 3 end subscript x.

big idea

When

and

the graph of f open parentheses x close parentheses equals b to the power of x

passes through the points open parentheses 0 comma space 1 close parentheses

and

open parentheses 1 comma space b close parentheses.

Then, the graph of its inverse, g open parentheses x close parentheses equals log subscript b x

, passes through the points open parentheses 1 comma space 0 close parentheses

and

open parentheses b comma space 1 close parentheses.

This means that log subscript b 1 equals 0

and

for all values of b such that b greater than 0

and

In summary, here are the possible shapes of g open parentheses x close parentheses equals log subscript b x.

big idea

When

and

the domain of f open parentheses x close parentheses equals log subscript b x

open parentheses 0 comma space infinity close parentheses

and its range is open parentheses short dash infinity comma space infinity close parentheses.

Also note the following:

When the function is decreasing on its domain.
When the function is increasing on its domain.

Since the graph of a logarithmic function passes the horizontal line test, logarithmic functions are also one-to-one. Then, logarithmic functions also have the inverse property.

The one-to-one property for logarithmic functions tells us that if two logarithmic expressions with the same base are equal to each other, then the arguments must also be equal. It also tells us that if two quantities are equal, their logarithms in the same base are also equal. This property will be used more frequently when solving equations.

term to know

One-to-One Property for Logarithmic Functions

Consider the function f open parentheses x close parentheses equals log subscript b x comma

where

and

Then, for quantities R and S, log subscript b R equals log subscript b S

if and only if R equals S.

3. Comparing Graphs of Logarithmic Functions

Now that we know what the graph of a logarithmic function looks like, we can observe relationships between different graphs.

First, let’s compare two logarithmic graphs with different bases.

EXAMPLE

In this example, we’ll graph f open parentheses x close parentheses equals log subscript 2 x

and

g open parentheses x close parentheses equals log subscript 5 x

on the same pair of axes.

First, here is the table of values for each function:

				1	2	4	8
	-3	-2	-1	0	1	2	3

				1	5	25	125
	-3	-2	-1	0	1	2	3

In the figure below, you see the graphs of the functions. The thinner graph is f open parentheses x close parentheses equals log subscript 2 x

and the thicker graph is g open parentheses x close parentheses equals log subscript 5 x.

A graph with an x-axis ranging from 0 to 16 and a y-axis ranging from −4 to 4 in increments of 2. The graph has two curves, where one curve with the function g of x equals log base 5 of x rises along the negative y-axis, opens downward, and extends into the first quadrant, passing through the points (1 over 5, -1), (1, 0) and (5, 1). The other curve with the function f of x equals = log base 2 of x rises along the negative y-axis, opens downward, and extends into the first quadrant, passing through the points at (1 over 2, -1), (1, 0) and (2, 1).

Observe the following:

Both graphs contain the point and have vertical asymptote
When the graph of g is above the graph of f.
When the graph of f is above the graph of g.

In general, considering the graphs as x rightwards arrow infinity comma

the logarithmic function with the smaller base will be above the graph of the logarithmic function with the larger base. Later in this course, we will actually see that the graphs of f and g are constant multiples of one another.

try it

Consider the graphs marked A and B in the figure. One of the graphs is y equals log x

and the other is y equals log subscript 3 x.

A graph with an x-axis and a y-axis. The graph has two curves labeled ‘A’ and ‘B’ that rise along the negative y-axis in the lower right quadrant, open downward, pass through (1, 0), and extend into the first quadrant. To the right of (1, 0), curve A runs higher than curve B, which is closer to the positive x-axis.

Which function is represented by each graph?

Graph A is y equals log subscript 3 x

and Graph B is y equals log x.

Let’s now do an exploration where we compare two logarithmic functions, but this time, the bases are reciprocals of each other.

EXAMPLE

Consider the functions f open parentheses x close parentheses equals log subscript 3 x

and

The table below shows their values for selected values of x.

			1	3	9	27
-3	-2	-1	0	1	2	3
3	2	1	0	-1	-2	-3

For a given x-value, notice that the output values are opposites. What does this mean for the graphs?

The graphs are shown below, with selected points plotted. Notice that both graphs contain the point open parentheses 1 comma space 0 close parentheses.

A graph with an x-axis ranging from −2 to 24 in increments of 2 and a y-axis ranging from −4 to 4. The graph has two curves, where one curve with the function g of x equals log base (1 over 3) of x, descends along the y-axis, passes through the marked points at (1 over 3, 1), (1, 0), and (9, −2), and extends into the fourth quadrant. The other curve with the function f of x equals log base 3 of x, rises along the negative y-axis, opens downward, and extends into the first quadrant, passing through the marked points at (1 over 3, −1), (1, 0), and (9, 2). Both the curves intersect at the marked point (1, 0).

As it turns out, each graph is a reflection of the other over the x-axis!

This means that log subscript 1 divided by 3 end subscript x equals short dash log subscript 3 x.

log subscript 1 divided by 3 end subscript x equals short dash log subscript 3 x.

As a result of the last example, we have the following.

formula to know

Logarithms With Reciprocal Bases

For any base b greater than 1 comma

log subscript 1 divided by b end subscript x equals short dash log subscript b x.

This result is important since it establishes a correspondence between logarithms with bases that are reciprocals of each other. Going forward, you’ll notice that all applications seen in this course (and in future courses) use bases that are larger than 1. This is because the common and natural logarithms both use bases that are larger than 1.

try it

Consider the functions f open parentheses x close parentheses equals log subscript 6 x

and

g open parentheses x close parentheses equals log subscript 1 divided by 6 end subscript x.

Write g (x ) in terms of f (x ).

log subscript 1 divided by 6 end subscript x equals short dash log subscript 6 x comma

g open parentheses x close parentheses equals short dash f open parentheses x close parentheses

summary

In this lesson, you learned that the graphs of logarithmic functions are obtained by using the fact that logarithmic functions are inverses of exponential functions. Since there are two basic shapes of exponential graphs, there are also two basic shapes of logarithmic graphs: one corresponding to the graph of a logarithmic function where b > 1 and the other corresponding to the graph of a logarithmic function where 0 < b < 1. You also learned that upon further comparison of the graphs of logarithmic functions, the graphs of y equals log subscript b x

and

y equals log subscript 1 divided by b end subscript x

are reflections of each other over the x-axis. Because of this fact, we focus on logarithmic functions with bases that are greater than 1. This way, we can still reference the natural and common logarithms.

SOURCE: THIS TUTORIAL HAS BEEN ADAPTED FROM OPENSTAX "PRECALCULUS” BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/DETAILS/BOOKS/PRECALCULUS-2E. LICENSE: CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL.

Terms to Know

One-to-One Property for Logarithmic Functions: Consider the function $f open parentheses x close parentheses equals log subscript b x comma$ where $b greater than 0$ and $b not equal to 1.$ Then, for quantities R and S, $log subscript b R equals log subscript b S$ if and only if $R equals S.$

Formulas to Know

Logarithms With Reciprocal Bases: For any base $b greater than 1 comma$ $log subscript 1 divided by b end subscript x equals short dash log subscript b x.$

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Graphs of Logarithmic Functions

Table of Contents

1. The Graph of a Logarithmic Function Where b > 1

2. The Graph of a Logarithmic Function Where 0 < b < 1

3. Comparing Graphs of Logarithmic Functions