MODULE
16
STUDY GUIDE REVIEW
Logarithmic Properties and
Exponential Equations
Study Guide Review
Essential Question: How do the properties of logarithms allow
you to solve real-world problems?
ASSESSMENT AND INTERVENTION
KEY EXAMPLE
Simplify: log 5 5
x+2
MODULE
16
Key Vocabulary
exponential equation
(ecuación exponencial)
(Lesson 16.1)
+ log 2 16 .
3
Apply properties of logarithms.
log 5 5 x + 2 + log 2 16 3 = (x + 2)log 5 5 + 3log 2 16
Assign or customize module reviews.
= (x + 2) + 3log 2 (2 4)
= x + 2 + 3(4)
= x + 14
MODULE
PERFORMANCE TASK
KEY EXAMPLE
(Lesson 16.2)
Solve the equation: 4 3x + 1 = 6.
COMMON
CORE
4 3x + 1 = 6
log4 3x + 1 = log6
Mathematical Practices: MP.1, MP.2, MP.4, MP.6, MP.7
F-BF.5, F-LE.4
© Houghton Mifflin Harcourt Publishing Company
• What is the formula for radiocarbon dating?
Encourage students to derive this formula by
solving the half-life formula for t. The formula is
Bring down the exponent.
log6
3x + 1 = _
log4
log6
3x = _ - 1
log4
SUPPORTING STUDENT REASONING
Students should begin by thinking about how to find
a model for the age of a carbon-containing material.
They can then do research, or you can provide them
with specific information. Here is some of the
information they may ask for.
Take the log of both sides.
(3x + 1)log4 = log6
(
Rearrange to isolate x.
)
log6
1 _
- 1 ≈ 0.0975
x=_
3 log4
t
__
N = N 0(0.5) , where N is the amount of C-14
currently in the sample, N 0 is the original amount
of C-14, t __1 is the half-life of C-14, and t is the
2
time in years.
t __1
2
• What is the half–life of carbon-14? 5,730 years
Module 16
813
Study Guide Review
SCAFFOLDING SUPPORT
A2_MNLESE385900_U6M16MC 813
• Remind students of the properties of logarithms:
log a (u ⋅ v) = log a (u) + log a (v)
u
log a v = log a (u) - log a (v)
log a(u n) = nlog a(u)
log a(a b ) = b
_
• If students have trouble deriving the formula for carbon dating, you may wish
(N )
to provide it: t = t __1 × _____. The meanings of the variables are given above.
N
ln __
0
2
813
Module 16
ln(0.5)
3/29/14 2:22 AM
EXERCISES
Use properties of logarithms to simplify. (Lesson 16.1)
1.
log _3 0.216
4
5
log 4 4
2.
+ log 3 243
2
Assumptions
x+8
12
3.
SAMPLE SOLUTION
x-2
log 8 0.015625 x
• The ratio of carbon-12 to carbon-14 in the
atmosphere has been constant since the organism
from which the sample originated died.
log10 2x + 1 + log 3 9
4.
-2x
2x + 3
Method
Solve each equation. (Lesson 16.2)
5.
5 x = 50
6 x + 2 = 45
6.
log50
______
≈ 2.43
log45
______
-2 ≈ 0.125
log5
7.
20
2x + 3
= 15
(______ )
Starting from the half-life formula, derive the
formula for carbon dating by solving for t. Use the
properties of logarithms.
log6
3 5x + l = 150
8.
1 log15
_
- 3 ≈ -1.048
2 log20
(
_______
N = N 0 (0.5)
)
1 log150
_
- 1 ≈ 0.712
5 log3
N = (0.5)
___
t/t 1/2
t/t 1/2
N0
t ⋅ In (0.5)
N = ___
In ___
N 0 t _1
2
_N
In
N
t = t _ ⋅ _______
MODULE PERFORMANCE TASK
How Old Is That Bone?
22
_
The La Brea Tar Pits in Los Angeles contain one of the best preserved collections of Pleistocene
vertebrates, including over 660 species of organisms. An archeologist working at La Brea Tar
Pits wants to assess the age of a mastodon bone fragment she discovered. She measures that
the fragment has 22% as much carbon-14 as typical living tissue. Given that the half-life of
carbon-14 is 5370 years, what is the bone fragment’s age?
In
100
≈ 12, 500 years
t = 5730 ⋅ _______
814
In(0.5)
© Houghton Mifflin Harcourt Publishing Company
Start by listing in the space below the information you will need to solve the problem. Then
use your own paper to complete the task. Be sure to write down all your data and assumptions.
Then use graphs, numbers, words, or algebra to explain how you reached your conclusion.
Module 16
0
In(0.5)
Use this formula, with t __1 = 5730, N 0 = 100, and
2
N = 22, to calculate the age of the fragment:
1
2
Study Guide Review
DISCUSSION OPPORTUNITIES
A2_MNLESE385900_U6M16MC.indd 814
29/03/14 8:01 PM
• Do you think C-14 dating would be useful for finding the age of a dinosaur
fossil? Why or why not?
• Why can’t carbon dating be used to date stone tools?
Assessment Rubric
2 points: Student correctly solves the problem and explains his/her reasoning.
1 point: Student shows good understanding of the problem but does not fully
solve or explain.
0 points: Student does not demonstrate understanding of the problem.
Study Guide Review 814
Ready to Go On?
Ready to Go On?
ASSESS MASTERY
16.1–16.2 Logarithmic Properties and
Exponential Equations
Use the assessment on this page to determine if
students have mastered the concepts and standards
covered in this module.
• Online Homework
• Hints and Help
• Extra Practice
Use properties of logarithms to simplify. (Lesson 16.1)
log _6 2.0736 5
1.
2.
5
ASSESSMENT AND INTERVENTION
20
7
log 7 7 2x - 1 + log 3 81 2
3.
log 2 3.2 - log 2 0.025
4.
2x + 7
log 5 125 - log10 5x
-5x + 3
Solve each equation. Give the exact solution and an approximate solution to
three decimal places. (Lesson 16.2)
Access Ready to Go On? assessment online, and
receive instant scoring, feedback, and customized
intervention or enrichment.
7 2x = 30
5.
log30
_
≈ 0.874
2 0.5x + 7 = 215
ADDITIONAL RESOURCES
Differentiated Instruction Resources
• Reading Strategies
• Success for English Learners
• Challenge Worksheets
Assessment Resources
© Houghton Mifflin Harcourt Publishing Company
• Reteach Worksheets
2
5 2x - 1 = 20
0.5
2log7
7.
Response to Intervention Resources
6.
8.
(_ - 7) ≈ 1.496
log215
log2
(_ + 1) ≈ 1.431
log20
log5
10 3x - 3 = 15
log15
_1 (_
+ 3) ≈ 1.392
3
log10
ESSENTIAL QUESTION
How do you solve an exponential equation algebraically?
9.
Possible answer: Begin by taking the logarithm of both sides. This
allows you to use the Power Property of Logarithms to bring down
the variable, putting it in front of the logarithm. Then, use arithmetic
operations to isolate the variable in the normal way. At the end of the
process, you will likely need to use a calculator to find the value of a
logarithmic expression.
• Leveled Module Quizzes
Module 16
COMMON
CORE
A2_MNLESE385900_U6M16MC 815
815
Module 16
Study Guide Review
815
Common Core Standards
3/29/14 2:22 AM
Content Standards Mathematical Practices
Lesson
Items
16.1
1
F-BF.5
MP.7
16.1
2
F-BF.5
MP.7
16.1
3
F-BF.5
MP.7
16.1
4
F-BF.5
MP.7
16.2
5–8
F-LE.4
MP.2
MODULE
MODULE 16
MIXED REVIEW
MIXED REVIEW
Assessment Readiness
Assessment Readiness
1. For each function below, determine if the function has an inverse defined for all real
numbers. Select Yes or No for A–C.
A. f(x) = 4x 3 - 1
Yes
No
―
B. f(x) = √3x + 2
C. f(x) = 4x 2 + 2
Yes
Yes
ASSESSMENT AND INTERVENTION
No
No
2. Consider the equation 8 x+1 = 12. Choose True or False for each statement.
A. After bringing down the exponent,
the equation is (x + 1)log8 = log12.
True
False
B. The equation cannot be solved because
the bases are not the same.
True
False
True
False
C. The approximate value of x is 0.195.
Assign ready-made or customized practice tests to
prepare students for high-stakes tests.
3. At a constant temperature, the pressure, P, of an enclosed gas is inversely proportional
to the volume, V, of the gas. If P = 50 pounds per square inch when V = 30 cubic inches,
how can you find the pressure when the volume is 125 cubic inches?
k
Possible answer: The formula y = _
x can be used.kYou can assume P is x, and
, so k = 1500. From there,
V is y. That would make the first equation 30 = _
50
1500
125 = _
x , so x = 12. The amount of pressure needed is 12 pounds per
square inch.
ADDITIONAL RESOURCES
Assessment Resources
• Leveled Module Quizzes: Modified, B
4. A = P (1 + r) gives amount A in an account after n years after an initial investment P
that earns interest at an annual rate r. How long will it take for $250 to increase to $500
at 4% annual interest? Explain how you got your answer.
n
log 2
_
=n
log 1.04
17.67 ≈ n
COMMON
CORE
AVOID COMMON ERRORS
Item 1 Some students will miss that the item is
asking about the inverse of each function. Remind
students to read the entire question carefully, and
highlight or underline key words.
© Houghton Mifflin Harcourt Publishing Company
About 17.67 years; Possible explanation: Substitute, then use logarithms to
solve for n:
n
500 = 250(1.04)
n
2 = 1.04
log 2 = nlog1.04
Module 16
16
Study Guide Review
816
Common Core Standards
A2_MNLESE385900_U6M16MC 816
8/20/14 9:21 PM
Content Standards Mathematical Practices
Lesson
Items
10.1, 15.1
1*
F-BF.5, F-BF.A
MP.1
16.2
2
F-LE.4
MP.2
8.1
3*
N-Q.2
MP.6
16.2
4
F-LE.4
MP.6
* Item integrates mixed review concepts from previous modules or a previous course.
Study Guide Review 816