1
Algebra II: Strand 6. Exponentials and Logarithms; Topic 1. Characteristics of Exponentials and
Logarithms; Task 6.1.3
TASK 6.1.3: PROPERTIES OF LOGARITHMS
Solutions
1. Using the approximation methods from Task 6.1.2, fill in the table below, rounding to the
nearest thousandth. Note that the notation g(x) = log b x is used to denote the inverse
x
function of the exponential function f (x) = b .
If
3x = 36
then
x!
3.262
3.262
If
3x = 50
then
x!
log3 36 =
3.561
log3 50 =
3.561
If
3x = 3 then
If
3x = 125 then
If
3x = 220 then
x !1
log3 3=
x ! 4.395
x ! 4.91
1
log3 125 =
4.395
log3 220 =
4.91
2. Use your work from Exercise 1 and Task 6.1.2 to complete the following table:
x
log3 x
log10 x
Ratio:
log10 x
log 3 x
36
3.262
1.556
.477
50
3.561
1.699
.477
3
1
.477
.477
125
4.395
2.097
.477
220
4.91
2.342
.477
a. Explain what you discovered from the table. The ratio
log10 x
is constant. Or, log10 x
log 3 x
and log 3 x are directly proportional.
b. Examine the table again and write an equation that relates the three quantities log10 x ,
log 3 x , and log10 3 . Generalize your findings for any base b. Find log5 62 .
Participants will discover that
log10 x
log10 x
= log10 3 or
= log 3 x . The generalization
log 3 x
log10 3
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
2
Algebra II: Strand 6. Exponentials and Logarithms; Topic 1. Characteristics of Exponentials and
Logarithms; Task 6.1.3
will be the change of base formula for logarithms. They should then be able to use this to
log10 62
compute log5 62 =
! 2.564
log10 5
3. Given the graphs of
and
( )
s(x) = 1.5
x
and
x
( ) , sketch s!1 ( x ) = log1.5 x
v(x) = 1.3
()
v !1 x = log1.3 x . Use your sketch to explain why log1.5 2 ! log1.3 2 .
Generalize your conclusion. The filled in sketch below would be an acceptable answer with a
written explanation. When participants sketch the inverse functions, it becomes apparent (see
the line x=2 in the sketch below) that the two values are not equal (the graphs would have to
intersect there).
4. Fill in the following tables. Note any patterns that arise. Make a conjecture about general rules
on logarithms. Participants will observe the following properties readily from their table:
log b c x = x log b c , log b c ! d = log b c + log b d . The leader may have to prompt them to notice
(
)
other properties such as log ( b) = 1 , log ( b ) = s , and log (1) = 0 . As an extension and if time
s
b
b
b
permits, the leader can ask the participants to verify this graphically for a particular example and
to decide what implications this has on the properties of exponential functions.
Math notes
This task provides an exploratory exercise for discovering the properties of logs.
Teaching notes
Participants should work in groups of 2 or 3 on this task. Pass out blank transparencies so that
students can present their findings in part 4 to the class. Participants may be encouraged to enter
their data in lists for Exercise 4.
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
3
Algebra II: Strand 6. Exponentials and Logarithms; Topic 1. Characteristics of Exponentials and
Logarithms; Task 6.1.3
TASK 6.1.3: PROPERTIES OF LOGARITHMS
1. Using the approximation methods from Task 6.1.2, fill in the table below to the nearest
thousandth. Note that the notation g(x) = log b x is used to denote the inverse function of the
x
exponential function f (x) = b .
If
3x = 36
then
x!
If
3x = 50
then
x!
If
3x = 3 then
If
3x = 125 then
If
3x = 220 then
log3 36 =
log3 50 =
x!
log3 3=
x!
log3 125 =
x!
log3 220 =
2. Use your work from Exercise 1 and Task 6.1.2 to complete the following table:
x
log3 x
log10 x
Ratio:
log10 x
log 3 x
36
50
3
125
220
a. Explain what you discovered from the table.
b. Examine the table again and write an equation that relates the three quantities log10 x ,
log 3 x , and log10 3 . Generalize your findings for any base b. Find log5 62 .
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
4
Algebra II: Strand 6. Exponentials and Logarithms; Topic 1. Characteristics of Exponentials and
Logarithms; Task 6.1.3
( )
3. Given the graphs of s(x) = 1.5
v
!1
( x ) = log
1.3
x
( )
and v(x) = 1.3
x
, sketch s
!1
( x ) = log
1.5
x and
x . Use your sketch to explain why log1.5 2 ! log1.3 2 . Generalize your
conclusion.
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
5
Algebra II: Strand 6. Exponentials and Logarithms; Topic 1. Characteristics of Exponentials and Logarithms; Task 6.1.3
4. Fill in the following tables. Note any patterns that arise. Make a conjecture about general rules on logarithms.
3x
x
36
50
3
125
220
3x = c ! d
6 !6
25! 2
3!1
5! 25
20 !11
x
log10 c
x ! log10 3
log10 3x
36
50
3
125
220
3x
()
( )
log10 3
log10 d
(
log10 c ! d
)
()
( )
log10 c + log10 d
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education
Coordinating Board.