NAME:__________________________________________
DATE:____________
Algebra 2: Lesson 11 – 2 Graphing Logarithmic Functions
Learning Goals
1. How do we graph a logarithmic function?
2. What is the relationship between an exponential and a logarithmic function?
3. How do we find the domain and range of a logarithmic function?
Standard Form of Exponential Functions:
y = bx
Example 1: On the grid below, graph the exponential function y = 2x and its inverse.
a) What is the equation of the inverse of y = 2x?
How would we solve it for y?
b) In what quadrants does the graph of the
exponential function y = 2x lie?
c) In what quadrants does the graph of its inverse
lie?
d) Is the graph of the inverse of y = 2x a function?
If so, what is its domain and range?
Example 2: On the grid below, graph the exponential function y = ex and its inverse.
a) What is the equation of the inverse of y = ex?
How would we solve it for y?
b) In what quadrants does the graph of the
exponential function y = ex lie?
c) In what quadrants does the graph of its inverse
lie?
d) Is the graph of the inverse of y = ex a function?
If so, what is its domain and range?
A function f(x) is an exponential function:
Its inverse f-1(x) is a logarithmic function:
f(x) = bx
or y = bx
f-1(x) = logbx or y = logbx
Domain =
Domain =
Range =
Range =
( ), for the
Example 3: Graph the points in the table for the functions ( )
( ) and ( )
given values. Then, sketch smooth curves through those points and answer the questions that follow.
( )
-team
( )
( )
-team
( )
( )
( )
a) What do the graphs indicate
about the domain of your
functions?
b) Describe the -intercepts of the
graphs.
c) Describe the intercepts of the graphs.
d) Find the coordinates of the point
on the graph with -value .
e) Describe the behavior of the
function as
.
f) Describe the end behavior of the
function as
.
g) Describe the range of your
function.
h) Does this function have any
relative maxima or minima?
Explain how you know.
i) For which values of is
( )
( )?
Key Features of the graph of









( )
The domain is the positive real numbers, and the range is all real
numbers.
The graphs all cross the -axis at ( ).
A point on the graph is always (b, 1).
None of the graphs intersect the -axis.
They have the same end behavior as
( )
, and they have
( )
the same behavior as
.
The functions all increase quickly for
, then increase more and
more slowly.
As the value of increases, the graph will flatten as
.
There are no relative maxima or minima.
Inverse of the exponential function
Example 4: Graph the points in the table for your assigned function ( )
( ),and ( )
( ). Then sketch smooth curves through those points, and answer the questions that follow.
-team
( )
-team
( )
( )
( )
( )
( )
a. What is the relationship between your graphs in the Example 3 and your graphs from this
exercise?
b. Why does this happen? Use the change of base formula to justify what you have observed in
part (a).
c.
For which values of
is
( )
( )
Conclusion: From what we have seen of these sets of graphs of functions, can we state the relationship
( ) and
( ), for
between the graphs of
?

If
, then the graphs of
( ) and
( ) are reflections of each other
across the -axis.
Key Features of the graph of
( )
 The graph crosses the -axis at ( ).
 The graph does not intersect the -axis.
 The graph passes through the point (
).
, the function values increase quickly; that is, ( )
, the function values continue to decrease; that is,
.
 There are no relative maxima or relative minima.
 Inverse of the exponential function
 As
 As
( )
.