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16 M O D U L E

Study Guide Review

ASSESSMENT AND INTERVENTION

Assign or customize module reviews.

MODULE PERFORMANCE TASK

COMMON CORE

Mathematical Practices: MP.1, MP.2, MP.4, MP.6, MP.7 F-BF.5, F-LE.4

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.

? What is the formula for radiocarbon dating?

Encourage students to derive this formula by

solving the half-life formula for t. The formula is

_t_

N

=

N

0(0.5)

,t_1_ 2

where

N

is

the

amount

of

C-14

currently in the sample, N0 is the original amount

of C-14, t_1_ is the half-life of C-14, and t is the 2

time in years.

? What is the half?life of carbon-14? 5,730 years

? Houghton Mifflin Harcourt Publishing Company

STUDY GUIDE REVIEW

Logarithmic Properties and Exponential Equations

Essential Question: How do the properties of logarithms allow you to solve real-world problems?

KEY EXAMPLE

(Lesson 16.1)

Simplify: log5 5x + 2 + log2 163.

Apply properties of logarithms.

log5 5x + 2 + log2 163 = (x + 2)log5 5 + 3log2 16 = (x + 2) + 3log2 (24)

= x + 2 + 3(4) = x + 14

KEY EXAMPLE

(Lesson 16.2)

Solve the equation: 43x + 1 = 6.

43x + 1 = 6

log43x + 1 = log6

Take the log of both sides.

(3x + 1)log4 = log6

Bring down the exponent.

3x

+

1

=

_ log6 log4

3x

=

_ log6 log4

-

1

Rearrange to isolate x.

( ) x

=

_ 1 3

_ log6 log4

-

1

0.0975

MODULE

16

Key Vocabulary

exponential equation (ecuaci?n exponencial)

Module 16

813

Study Guide Review

SCAFFOLDING SUPPORT A2_MNLESE385900_U6M16MC 813

? Remind students of the properties of logarithms:

_ loga (u v) = loga (u) + loga (v)

log a

u v

=

log a

(u)

-

log a

(v)

loga(un) = nloga(u)

loga(ab ) = b

? If students have trouble deriving the formula for carbon dating, you may wish

( ) to

provide

it:

t

=

t

_1_ 2

?

_llnn_(_0_NN.__50 _) .

The

meanings

of

the

variables

are

given

above.

3/29/14 2:22 AM

813 Module 16

EXERCISES Use properties of logarithms to simplify. (Lesson 16.1)

1. log_3 0.2164 5 12

3. log8 0.015625x -2x

2. log4 4x - 2 + log3 2432 x + 8

4. log102x + 1 + log3 9 2x + 3

Solve each equation. (Lesson 16.2)

5. 5x = 50

_lo__g_5_0_

log5

2.43

7. 202x + 3 = 15

( ) _1

2

_lo__g_1_5_

log20

-

3

-1.048

6. 6x + 2 = 45

_lo__g_4_5_

log6

-2

0.125

8. 35x + l = 150

( ) _1

5

_lo__g_1_5_0_ log3

-

1

0.712

MODULE PERFORMANCE TASK

How Old Is That Bone?

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?

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.

SAMPLE SOLUTION

Assumptions

? The ratio of carbon-12 to carbon-14 in the atmosphere has been constant since the organism from which the sample originated died.

Method

Starting from the half-life formula, derive the formula for carbon dating by solving for t. Use the properties of logarithms.

N = N0 (0.5)t/t1/2

___N

N 0

=

( ) 0.5 t/t1/2

In

_N__

N 0

=

__t _

t_1

In

(0.5)

_ 2

t

=

t

_ 1

2

__In__N_N_0 _

In(0.5)

Use this formula, with t_1_ = 5730, N0 = 100, and 2

N = 22, to calculate the age of the fragment:

_ t = 5730 _I_n__12_02_0_ 12, 500 years In(0.5)

? Houghton Mifflin Harcourt Publishing Company

Module 16

814

Study Guide Review

DISCUSSION OPPORTUNITIES A2_MNLESE385900_U6M16MC.indd 814

? 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?

29/03/14 8:01 PM

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?

ASSESS MASTERY

Use the assessment on this page to determine if students have mastered the concepts and standards covered in this module.

ASSESSMENT AND INTERVENTION

Access Ready to Go On? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.

ADDITIONAL RESOURCES

Response to Intervention Resources ? Reteach Worksheets

Differentiated Instruction Resources ? Reading Strategies ? Success for English Learners ? Challenge Worksheets

Assessment Resources ? Leveled Module Quizzes

? Houghton Mifflin Harcourt Publishing Company

Ready to Go On?

16.1?16.2 Logarithmic Properties and Exponential Equations

Use properties of logarithms to simplify. (Lesson 16.1)

1. log_6 2.07365 5 20

2. log2 3.2 - log2 0.025 7

? Online Homework ? Hints and Help ? Extra Practice

3. log7 72x - 1 + log3 812 2x + 7

4. log5 125 - log105x -5x + 3

Solve each equation. Give the exact solution and an approximate solution to three decimal places. (Lesson 16.2)

5. 72x = 30

_ log30

2log7

0.874

6. 52x - 1 = 20

(_ ) log20

0.5

+ 1 1.431

log5

7. 20.5x + 7 = 215

(_ ) 2

log215 log2

-

7

1.496

8. 103x - 3 = 15

_ (_ ) 1

3

log15 log10

+

3

1.392

ESSENTIAL QUESTION

9. How do you solve an exponential equation algebraically?

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.

815 Module 16

Module 16

815

Study Guide Review

COMMON CORE

Common Core Standards

A2_MNLESE385900_U6M16MC 815

Lesson Items Content Standards Mathematical Practices

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

3/29/14 2:22 AM

MODULE 16 MIXED REVIEW

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) = 4x3 - 1

B. f(x) = 3x + 2

Yes

No

Yes

No

C. f(x) = 4x2 + 2

Yes

No

2. Consider the equation 8x+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

C. The approximate value of x is 0.195.

True

False

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?

PV12ois5ssy=i.bTl_ e1h5aax0tn0ws,woseuorlx:dT=mhea1k2foe.rTtmhheeulafaimrysot=uenq_kxtucoaatfinporbnee3ssu0us=reed_ 5n.k0Ye,oesudoeckdan=isa11s52su0p0mo.euFnrPodimss xpt,hearenred, square inch.

4. A = P (1 + r)n 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.

About 17.67 years; Possible explanation: Substitute, then use logarithms to solve for n:

500 = 250(1.04)n 2 = 1.04n

log 2 = nlog1.04

_ log 2 = n

log 1.04

17.67 n

16 M O D U L E

MIXED REVIEW

Assessment Readiness

ASSESSMENT AND INTERVENTION

Assign ready-made or customized practice tests to prepare students for high-stakes tests.

ADDITIONAL RESOURCES

Assessment Resources ? Leveled Module Quizzes: Modified, B

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

Module 16

816

Study Guide Review

COMMON CORE

Common Core Standards

A2_MNLESE385900_U6M16MC 816

Lesson Items Content Standards Mathematical Practices

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.

8/20/14 9:21 PM

Study Guide Review 816

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