Name________________________________
Date__________
Class__________
Exploring the Graph of Logarithmic Functions
Graph: ! y = log 3 x
1)
First, use the definition of logarithms to rewrite the function y = log 3 x as !3 y = x .
!
This is equivalent to !x = 3 y
Then, graph !x = 3 y by filling in the table below. (Since x is solved for instead of y, fill in the
table “backwards”.)
Graph ________
!x = 3
y
x
show work here
y
-2
-1
0
1
2
Plot the points from
the table onto the
coordinate plane.
Then draw a smooth
curve through the
points. Note, your
graph should get
closer and closer to
the y-axis but never
actually touch it!
Graph: y = log 2 x
!
2)
Rewrite the function y = log 2 x as _______________. This is equivalent to _______________.
!
(See #1 above for help.)
Graph ____________
x
show work here
y
-2
-1
0
1
2
Page 1 of 9
Plot the points from
the table onto the
coordinate plane.
Then draw a smooth
curve through the
points. Again, your
graph should get
closer and closer to
the y-axis but never
actually touch it!
©Let’s Algebra 2014
Graph: y = log 7 x
!
3)
Rewrite the function y = log 7 x as _______________. This is equivalent to _______________.
!
(See #1 on the previous page for help.)
Graph ____________
x
show work here
y
-1
0
1
Plot the points from
the table onto the
coordinate plane.
Then draw a smooth
curve through the
points. Again, your
graph should get
closer and closer to
the y-axis but never
actually touch it!
Please read the following observation note below!
If you look back at the logarithmic graphs in #1, #2, and #3, it is important that you observe the following:
!
y = log 3 x
(3, 1) is on the graph
x – intercept = 1
!
y = log 2 x
(2, 1) is on the graph
x – intercept = 1
!
y = log x
7
(7, 1) is on the graph
x – intercept = 1
In other words, the x-intercepts of all three graphs are 1. And they all contain the point (b, 1) where b is
the base of the logarithm. This will be true for any logarithm written in the form above where b > 0, b ≠ 1.
Therefore, you can graph logarithms much more quickly than you did in #1-3. All you need to do is locate
the x-intercept of 1 and then locate the point (b, 1), where b is the base of the logarithm. Once you have
located these two points, simply plot them and then draw a curve through the two points. Then you are
done!
Please continue to the next page.
Page 2 of 9
©Let’s Algebra 2014
4)
This is done as an example for you based on the
observation note from the previous page.
(9, 1)
Graph: ! y = log 9 x
The x – intercept = 1
The x – intercept = 1 and the point (9, 1) is on the line.
5)
Graph the following logarithmic function using the “new method” (Refer to #4 above and the
observation note on the previous page, if needed.)
Graph: y = log 5 x
!
The x-intercept = __________ and the point (
,
) is on the line.
For problems, #6-8, please refer to the graph above.
6)
Describe the domain and the range of the function y = log 5 x . Recall that the domain is the possible
!
values for the input (x) of a function, and the range is the possible values for the output (y).
7)
Identify the x-intercept and y-intercept of the function y = log 5 x . Recall that the x-intercept
!
is the x coordinate of a point where a graph crosses the x-axis. The y-intercept is the
y-coordinate of a point where a graph crosses the y-axis.
8)
Identify the asymptote of the function y = log 5 x . Recall that the asymptote is a line that a
!
graph approaches more and more closely, but never actually reaches.
Page 3 of 9
©Let’s Algebra 2014
You will now explore how to transform the parent function ! y = log 5 x
How does the graph y = log5 x compare to the to the “new” graph y = log 5(x –3)+ 2 ?
!
Recall that the –3 translates the parent function right 3 units and the + 2 translates the parent
function up 2 units. (You may refer to the last page of this packet for the rules of transformations.)
9)
Graph the following two logarithmic functions on the same coordinate plane. Use different
colored pencils to graph each function.
y = log 5 x
!
The x–intercept = _____________
The point (
,
) is on the line.
• Note: You already graphed this
function on the previous page!
!
y = log 5(x –3)+ 2
Hint:
• 
• 
• 
• 
Translate the asymptote x = 0 right 3 units.
Translate the point (1, 0) right 3 units and up 2
units.
Translate the point (5, 1) right 3 units and up 2 units.
Draw a smooth curve through the new points.
There will be two
graphs on this
coordinate plane!
The domain of y = log 5(x –3)+ 2 is ___________________.
!
The range of y = log 5(x –3)+ 2 is _____________________.
!
The asymptote of y = log 5(x –3)+ 2 is __________________.
!
Page 4 of 9
©Let’s Algebra 2014
Exercises:
For exercises a-c, graph the each logarithmic function. Describe the domain and range.
Identify the x-intercept, the y-intercept, and the asymptote. Remember to use the “new
method” when graphing. (See the observation note on the bottom of page 2 and #4 on
the top of page 3 for guidance.)
a)
Graph: y = log 8 x
!
The x – intercept = _______ and the point (
,
) is on the line
The domain of ! y = log 8 x is________________.
The range of ! y = log 8 x is _________________.
The x – intercept of y = log 8 x is _____________.
!
The y – intercept of y = log 8 x is ____________.
!
The asymptote of y = log 8 x is _______________.
!
b)
Graph: y = log 4 x
!
The x – intercept = _______ and the point (
,
) is on the line
The domain of of y = log 4 x is______________.
!
The range of y = log 4 x is _________________.
!
The x – intercept of ! y = log 4 x is ____________.
The y – intercept of y = log 4 x is ____________.
!
The asymptote of y = log 4 x is ______________.
!
Page 5 of 9
©Let’s Algebra 2014
Graph: ! y = log x
Hint: the common logarithm has a base of 10
c)
The x – intercept = _______ and the point (
,
) is on the line
The domain of of ! y = log x is______________.
The range of ! y = log x is _________________.
The x – intercept of ! y = log x is ____________.
The y – intercept of ! y = log x is ____________.
The asymptote of ! y = log x is ______________.
For exercises d-e, graph the parent function first. Then, on the same coordinate plane, graph the
“new” function by transforming the parent function. Use different colored pencils for each graph.
d)
parent function
!
y = log 6 x
The x-intercept is __________.
The point (
,
) is on the graph
“new” function
!
y = log 6 (x − 2)− 3
Hint:
The asymptote x = 0 is translated right 2.
units. The two points on the parent function
are translated right 2 units and down 3 units.
The domain of the “new” function is __________.
The range of the “new” function is ____________.
The asymptote of the “new” function is ________.
Page 6 of 9
©Let’s Algebra 2014
e)
“new” function
parent function
!
y = log 8 x
!
The x-intercept is __________.
The point (
,
) is on the graph
y = − log 8 (x + 4)−1
Hint:
The asymptote x = 0 is translated left
4 units. The two points on the parent
function are reflected over the x-axis.
These two points are then translated
left 4 units and down 1 unit.
The domain of the “new” function is __________.
The range of the “new” function is ____________.
The asymptote of the “new” function is ________.
Please continue to the next page…
Page 7 of 9
©Let’s Algebra 2014
Without graphing, describe the transformation from the parent function.
(Refer to the last page, p. 9, for extra help.)
f)
parent function:
“new” function:
y = log 2 x
y = log 2 (x ! 1)
Describe the transformation:
g)
parent function:
“new” function:
y = log 2 x
y = log 2 x ! 1
Describe the transformation:
h)
parent function:
“new” function:
y = log x
y = log(x ! 2) ! 5
Describe the transformation:
i)
parent function:
“new” function:
1
y = ! log 4 (x ! 1) + 2
2
y = log 4 x
Describe the transformation:
j)
parent function:
“new” function:
y = log 5 x
y = !2 log 4 (x + 5) + 3
Describe the transformation:
Page 8 of 9
©Let’s Algebra 2014
Reference guide for transforming logarithmic functions
“New” function
y = a log b (x ! h) + k
Page 9 of 9
•
If the absolute value of a is greater than one, a stretch occurs.
•
If the absolute value of a is greater than zero and less than 1, a compression
(shrink) occurs.
•
If a is less than zero a reflection occurs over the x – axis.
•
– h causes a translation right h units. This causes the asymptote to translate right h
units as well.
•
+ h causes a translation left h units. This causes the asymptote to translate left h
units as well.
•
+ k causes a translation up k units.
•
– k causes a translation down k units.
©Let’s Algebra 2014