1
Algebra II: Strand 6. Exponentials and Logarithms; Topic 1. Characteristics of Exponentials and
Logarithms; Task 6.1.2
TASK 6.1.2: WHAT IS X WHEN
y = bx ?
Solutions
Given below are the graphs of two exponential functions f and g.
1. Explain why f and g each have inverse functions. Both f and g satisfy the horizontal line
test or “when I reflect each function about the line y=x, I get a function without
restricting the domain of the original function.”
2. Carefully sketch the inverse functions of each on the plots above.
a. What is the domain of the inverse function of f? of g? Domain is x ! 0 for both
functions.
b. What is the range of the inverse function of f? of g? Range is all real numbers for
both functions.
3. For an exponential function h(x) = bx , b is called the base of the exponential function.
From the graphs of f and g given above, how would one determine the approximate value
of the base of each? Write plausible defining expressions for both f and g using this
approximate value of the base. Look at the value of f(1) and g(1). f (x) = (1.5) x and
g(x) =
()
2
3
x
.
()
4. Use your sketch to determine approximate values of f !1 (2) and g !1 2 where f !1 and
g !1 denote the inverse functions of f and g respectively. Explain how you determined
these values. Participants should take advantage of the symmetry. Ideally one wants to
see sketches similar to:
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 6.1.2
5. Explain why determining the value of f !1 (2) is equivalent to asking the question: “what
is x when f (x) = 2 ?” or “what is x when 2 = bx (where b is the base of the exponential
function f)?” Participants should refer to the symmetry in their graph or they could also
f !1 2
state that 2 = "# f o f !1 $% 2 = b ( )
6. Think about the following question: “What is x when 10 x = 36 ?” There are several ways
to investigate this question. We could begin by finding an interval that contains the
solution. That is, we can easily calculate 101 = 10 and 102 = 100 and this would help us
determine that the solution is between 1 and 2. Also, since 101.5 ! 31.6 we could say that
1.5<x<2. We can continue this process to determine an approximate value for x. Use this
method for estimating the approximate value of x to the nearest thousandth for the
following:
()
Estimates Using Guess-and-Check
10 x = 36
x ! 1.556
10 x = 50
x ! 1.699
10 x = 3
x ! .477
10 x = 125
x ! 2.097
10 x = 220
x ! 2.342
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 6.1.2
7. Determine a graphical method for obtaining estimates for the solutions. Explain your
method. For example, for the first entry participants may use the following:
8. Using the LOG key, find each of the following values to the nearest thousandth.
9.
log 36 =
1.556
log50 =
1.699
log 3 =
.477
log125 =
2.097
log 220 =
2.342
Compare the values in the tables from Exercises 6, 7, and 8.
a. Explain what the calculator displays when the LOG key is used.
For example, log 36 is the solution to 10 x = 36 . That is, log c is the solution to
10 x = c for a given c>0.
b. Without using your calculator, answer the following.
i. If 101.653 = 45 , what is log 45 ? 1.653
ii. If log132 = 2.121 , what is 102.121 ? 132
iii. log104 = 4
iv. 10log1000 = 1000
10. LOG on your calculator is short for common logarithm. Give the exponential function for
which the common logarithm function is the inverse function. Explain your reasoning.
Based upon the tables for powers of 10 and the relationship correspondence with the
tables for LOG we conclude that the base of the exponential function for which LOG is
the inverse function is 10. Observe that in iii and iv of Exercise 9 it is easier to see that if
we compose the two functions we get the identity function.
11. Can a common logarithm of a real number be negative? If so, give an example. If not,
explain why not. Yes, if the number is between 0 and 1. Encourage participants to see
this graphically.
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 6.1.2
12. Do negative numbers have common logarithms? If so, give an example. If not explain
why not. No, the domain of the common logarithm is equivalent to the range of the
exponential function, which is 0,!
( )
Math notes
The approach developed here emphasizes the fact that exponential functions of base b and
logarithm functions of base b are inverse functions.
Teaching notes
This task can be done with little direct coaching by the leader. It is important to circulate as
the students work in groups to ensure that they are using a qualitative approach (rather than
button-pushing) to working the exercises. For example, in Exercise 4 ask participants to
show how they obtain their solution graphically. In Exercise 6, the process they are using is a
bisection method. The leader may have to clarify the next step of the process (that is choose
the midpoint of the interval that contains x) if participants do not develop this on their own.
Technology notes
In Exercise 7, participants need to be reminded to choose a friendly viewing window. Also,
the leader may need to review how to find the intersection of two curves on the graphing
calculator.
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 6.1.2
TASK 6.1.2: WHAT IS X WHEN y = bx?
Given below are the graphs of two exponential functions f and g.
1. Explain why f and g each have inverse functions.
2. Carefully sketch the inverse functions of each on the plots above.
a. What is the domain of the inverse function of f? of g?
b. What is the range of the inverse function of f? of g?
x
3. For an exponential function h(x) = b , b is called the base of the exponential function.
From the graphs of f and g given above, how would one determine the approximate value
of the base of each? Write plausible defining expressions for both f and g using this
approximate value of the base.
4. Use your sketch to determine approximate values of f
g
!1
( 2 ) and g ( 2 ) where
!1
f
!1
and
!1
denote the inverse functions of f and g respectively. Explain how you determined
these values.
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.
6
Algebra II: Strand 6. Exponentials and Logarithms; Topic 1. Characteristics of Exponentials and
Logarithms; Task 6.1.2
5. Explain why determining the value of f
!1
( 2 ) is equivalent to asking the question:
x
“what is x when f (x) = 2 ?” or “what is x when 2 = b (where b is the base of the
exponential function f)?”
x
6. Think about the following question: “What is x when 10 = 36 ?” There are several
ways to investigate this question. We could begin by finding an interval that contains the
1
2
solution. That is, we can easily calculate 10 = 10 and 10 = 100 and this would help
1.5
us determine that the solution is between 1 and 2. Also, since 10 ! 31.6 we could say
that 1.5<x<2. We can continue this process to determine an approximate value for x. Use
this method for estimating the approximate value of x to the nearest thousandth for the
following:
Estimates Using Guess-and-Check
10 x
10 x
10 x
10 x
10 x
= 36
= 50
=3
= 125
= 220
x!
x!
x!
x!
x!
7. Determine a graphical method for obtaining estimates for the solutions. Explain your
method.
Estimates Using Graphing Technique
10 x
10 x
10 x
10 x
10 x
= 36
= 50
=3
= 125
= 220
x!
x!
x!
x!
x!
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.
7
Algebra II: Strand 6. Exponentials and Logarithms; Topic 1. Characteristics of Exponentials and
Logarithms; Task 6.1.2
8. Using the LOG key, find each of the following values to the nearest thousandth.
log 36 =
log 50 =
log 3 =
log125 =
log 220 =
9. Compare the values in the tables from Exercises 6, 7, and 8.
c. Explain what the calculator displays when the LOG key is used.
d. Without using your calculator, answer the following.
1.653
i. If 10
= 45 , what is log 45 ?
ii. If log132 = 2.121, what is 10
2.121
?
4
iii. log10 =
iv. 10
log1000
=
10. LOG on your calculator is short for common logarithm. Give the exponential function for
which the common logarithm function is the inverse function. Explain your reasoning.
11. Can a common logarithm of a real number be negative? If so, give an example. If not,
explain why not.
12. Do negative numbers have common logarithms? If so, give an example. If not explain
why not.
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.