Algebra 2 and Trigonometry - White Plains Middle School · Algebra 2 and Trigonometry Chapter 8:...

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Algebra 2 and

Trigonometry

Chapter 8: Logarithms

Name:______________________________

Teacher:____________________________

Pd: _______

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Table of Contents

Day 1: Chapter 8-1/8-2: Converting an Exponential Equation into a Logarithmic Equation SWBAT: Convert an Exponential Equation into a Logarithmic Equation Pgs. 3 – 11 in Packet

HW: 326 – 327 #3 – 73 (every other odd) Day 2: Chapter 8-3: Properties of Logs

SWBAT: Learn the properties of logs. Pgs. 12– 18 in Packet

HW: Page 331 – 332 #15 – 53 (every other odd) Day 3: Chapter 8-3: Review of Properties of Logs

SWBAT: Learn the properties of logs. “REVIEW DAY”

QUIZ on Day 4

Day 4: Chapter 8-6/8-7: Solving Equations with Logs

SWBAT: Solve Equations using Logs

Pgs. 19 – 23 in Packet

HW: Page 343 #3 – 14 Page 346 #3 – 17 (odd) Day 5: Chapter 8-5: The Natural Log

SWBAT: Solve Exponential Equations with like and unlike bases


HW: Pgs. 31 – 33 in Packet

Day 6: Review

SWBAT: Solve Log Equations


HW: Booklet (not in this Packet)

QUIZ on Day 7

Day 7: Exponential Growth and Decay

SWBAT: Solve applications of Exponential Equations



Day 8: More Applications of Exponential Equations

SWBAT: Solve applications of Exponential Equations


HW: _________________

Day 9: Inverses and Graphs of Logarithmic Functions

SWBAT: Graph Logarithmic Functions



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Name _________________ Date ______ Per. _____/Ms. Williams

Day 1 – Converting an Exponential Equation into a Logarithmic

Equation

SWBAT: Convert an Exponential Equation into a Logarithmic Equation

Warm - Up Solve the following system of equations algebraically:

33

99 22

yx

yx

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Concept 1: Converting from Exponential to Logarithmic Form and Vice Versa

Until now, there was no way to isolate y in an equation of the form . You cannot take the “yth root” of something if that something isn’t a value. The word “Logarithm” means “power.” When you see the function “log,” you should translate that into “the power I raise…” Example: Log2 8 = 3

Example: X = Log10 1000

So,

=

The power I raise 2 to to get

8 is 3 What 10000

The power I raise 10 to to get

is

5

OR

6

Concept 1: How do we change between log form and exponential form?

Teacher Modeled Student Try it!

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Concept 2: Simplifying a Log Expression

Rule:

Teacher Modeled Student Try it!

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Concept 3: How Do we evaluate Common Logs?

Base with NO Exponent Radical with NO index

X √ means……. means…….

Logarithms with a base Logarithms with NO base

means……. means…….

On the graphing Calculator

9

Evaluate each.

10

Challenge: Solve the following Equation

Summary/Closure:

Exit Ticket:

Homework: Page 326 – 327 #3 – 73 (every other odd)

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12

Name _________________ Date ______ Per. _____/Ms. Williams

Day 2 – Properties of Logs SWBAT: Learn the properties of Logs

Solve.

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Algebra2/Trig: Logarithm Distribution Rules Let M = 107 and N = 1012 . Then log M =______________ and log N = ________________

What is ? _________________________

(This means multiply M and N first, then determine the log of the product.)

What is (

)?__________

(This means divide M by N first, then determine the log of the quotient.)

Now, let M = 107 and c = 5. What is ____________? (This means raise M to the power of c first, then determine the log of that answer.)

Log Distribution Rules (Summary): Product Rule

Quotient Rule

(

)

Power Rule

Expanding Logarithms

Examples

1. log (a2b3c)= log a2 + log b3 + log c= (product rule) 2 log a + 3 log b + log c (power rule)

2. (

)=

log a2 + log b3 - log c5 = (product/quotient rule)

2 log a + 3 log b - 5log c (power rule)

3. (

) =

log a2 – (log b3 + log c5)= (product/quotient rule) 2 log a – (3 log b + 5log c) (power rule) 2 log a – 3 log b - 5log c (distribute)

4. ( √

√ ) =

(product/quotient rule)

(evaluate log 100)

** RECALL ROOTS ARE EXPONENTS

The log of a product is the ____________ of the individual logs.

The log of a quotient is the ____________ of the individual logs.

The log of a power : ____________ the power by the individual log.

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Concept 1: Expanding Logs Expand each using the properties of Logs 1. 2.

3. 4.

5. Fdf 6.

7. 8.

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Concept 2: Condensing Logs 9. 10.

11. 12.

13. 14.

Use the properties of logarithms and the logarithms provided to rewrite each logarithm in terms of the variables given.

15. 16.

16

13. 18.

Harder Problems Condense each expression to a single logarithm.

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Challenge

Summary/Closure:

Exit Ticket:

Homework: Page 331 – 332 #15 – 53 (every other odd)

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Day 4 –Solving Logarithmic Equations

SWBAT: Solve Equations involving Logs

Warm - Up

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Concept 1: Taking the log of both sides to solve an exponential equation (Can’t change to the same base) Solve for the variable in each problem. If there is some reason the base is not isolated before you begin, ISOLATE the base first before you take the log of both sides.

1.

2.

3.

4.

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Concept 2: Equations with more than one log on one side. If an equation has more than one “log,” use the rules for log distribution to CONDENSE the logs on a side, and then proceed as you would otherwise. Usually this will involve converting the equation to exponential form. 5. 6.

7. 8.

If there are logs on each side of the equation, _______________________

9. 10.

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CHALLENGE SUMMARY Exit Ticket Solve the equation. Round your answer to the nearest ten-thousandth.

Homework: Page 343 #3 – 14 Page 346 #3 – 17 (odd)

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Day 5 –The Natural Log

SWBAT: solve equations using natural logs.

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Rules for the Natural Log (Exactly the same as regular Logs)

Example:

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Inverses lnx and e

x are inverse functions – If you do one, then the other, you get

what you started with. In symbols,

Ex. lnx = 4 e

x = 12

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Rules to solve an exponential equation with base e:

1. Isolate base e first. 2. Take the natural log of both sides to ”bring down the exponent”.

3. Solve for the variable.

Rules to solve an logarithmic equation with base e:

1. Condense each equation if possible to put all ln terms together 2. Exponeniate both sides to remove the ln function.

3. Solve for the variable.

31. 32.

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33. 34.

Condense each equation if possible to put all log or ln terms together If a Log or natural log is equal to another log of the same base then their contents are =.

35. 36.

Challenge

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SUMMARY Exit Ticket Expand the natural logarithm.

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Day 5 – HW

32

33

34

35

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Day 7: Exponential Growth and Exponential Decay

SWBAT: Solve problems involving exponential growth, exponential decay

Warm-up:

Exponential growth occurs when a quantity increases by the same rate r in each period t.

When this happens, the value of the quantity at any given time can be calculated as a function of

the rate and the original amount.

Exponential decay occurs when a quantity decreases by the same rate r in each time period t.

Just like exponential growth, the value of the quantity at any given time can be calculated by

using the rate and the original amount.

Explain:

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Concept 1: Finding “new” Value

Example 1:

The original value of a painting is $9,000 and the value increases by 7% each year. Write an exponential growth

function to model this situation. Then find the painting’s value in 15 years.

Example 2: The population of a town is decreasing at a rate of 3% per year. In 2000 there were 1700 people. Write an

exponential decay function to model this situation, and then find the population in 2012.

Practice:

1) A sculpture is increasing in value at a rate of 8% per year, and its value in 2000 was $1200. Write an

exponential growth function to model this situation, and then find the sculpture’s value in 2006.

Answer: ______________

Step 1: Write the exponential growth function for this situation

Step 2: Find the value in 15 years.

Step 1: Write the exponential growth function for this situation

Step 2: Find the value in 2006.

y = _________ ( 1 ______ ) ____

y = _________ ( _____ ) ____

y = _________ ( 1 ______ ) ____

y = _________ ( _____ ) ____

Step 1 Write the exponential decay function for this situation

Step 2 Find the value in 12 years.

y = _________ ( 1 ______ ) ____

y = _________ ( _____ ) ____

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Example 3:

Is the equation A = 1500 (0.86)t

a model of exponential growth or exponential decay, and what is the rate

(percent) of change per time period?

1) exponential growth and 14%


3) exponential decay and 14%


Example 4:

Is the equation A = 5000 (1.04)t







Example 5:

The fish population of Lake Collins is decreasing at a rate of 4% per year. In 2002 there were about 1,250 fish.

Determine whether this model is an exponential growth or exponential decay, and which equation can be used

to find the population in 2008?

1) exponential growth ; y = 1250(0.96)6


3) exponential decay ; y = 1250(1.04)6


Example 6:

The value of a gold coin picturing the head of the Roman Emperor Vespasian is increasing at the rate of 5 per

year. The coin is worth $105 now. Determine whether this model is an exponential growth or exponential

decay, and which equation can be used to find what the coin will be worth in 11 years?




Explain here!

Explain here!

Explain here!

Explain here!

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Concept 2: Finding “Time”

Example 7: The number of employees at a certain company is 1440 and is increasing at a rate of 1.5% per year. If the

number of employees now is 1646, how long did it take for the company to reach this many employees.

1) The fish population in a local stream is decreasing at a rate of 3% per year. The original population was

48,000. If the number of fish now is 38,800, how long did it take for the fish to reach this population?

2) The deer population of a game preserve is decreasing by 2% per year. The original population was 1850. If

the number of deer now is 1700, how long did it take for the deer to reach this population?

_______ = ______ ( 1 ______ )

_______ = ______ ( 1 ______ )

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Regents Questions:

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Summary:

Exit Ticket:

1)

2) The value of a gold coin picturing the head of the Roman Emperor Marcus Aurelius is increasing at the rate

of 7 per year. If the coin is worth $145 now, what will it be worth in 14 years?

1) $308.44 3) $373.89

2) $287.10 4) $243.00

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Homework

Write an exponential growth function to model each situation. Then find the value of the function after

the given number of years.

1)

2)

3)

Write an exponential decay function to model each situation. Then find the value of the function after

the given number of years.

4)

5)

6)

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7) Is the equation A = 3200 (0.70)t







8) Is the equation A = 1756 (1.17)

t a model of exponential growth or exponential decay, and what is the rate






9) Is the equation A = 10,000 (0.45)t







10) Is the equation A = 5400 (1.07)t







Explain here!

Explain here!

Explain here!

Explain here!

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11)

12)

13)

\

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Day 8 - Application of Exponential Equations

All of the following formulae, should they appear on my exam or the regents, will be given to you. But you should know what they mean and how they are used. For ALL PROBLEMS, round to the nearest TENTH when necessary.

Application #1: Exponential Growth and Decay

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Example 1: Astatine-218 has a half-life of 2 seconds. This means that every 2 seconds, half of the radioactive material decays into non-radioactive material. Find the amount left from a 500 gram sample of astatine-218 after 10 seconds.

Example 2: A bacterial culture triples every P hours. If the culture started with 13000 bacteria, and there are

24000 after 2 hours, what is the value of P in hours? Example 3: A bacterial culture doubles every 5 hours. How long will it take for the culture to quadruple?

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You Try it! 1.

2.

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Application #2: Interest and Investments

Vocabulary:

Principal is the initial investment (or loan) amount.

Balance is the amount of money remaining on the loan or in your bank

Interest is the money that is earned when someone lends someone money. o Simple interest is interest calculated one time. o Compound interest is interest calculated a certain number of times a year. The

interest percent quoted is SPLIT UP into a certain number of equal parts, and then it’s compounded (calculated and paid out) that same number of times a year. You earn interest on the interest. That’s why your money grows exponentially.

Example 4: Write a compound interest function to model $1200 invested at a rate of 2% compounded quarterly. Then find the balance after 3 years. Example 5: Write a compound interest function to model $15,000 invested at a rate of 2.03% compounded semi-annually. How long will it take for the amount of investment to triple?

Final Value of Investment (

)

A: amount (balance) after t years P: Principal value t: number of years that has passed n: number of times per year interest is compounded r: annual interest rate expressed as a decimal

Compounding… Annually: ___ per year Bi-Annually/Semi-annually: ___ per year Quarterly: ___ per year Monthly: ___ per year Daily: ___ per year

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You Try it!

1. Write a compound interest function to model $15,000 invested at a rate of 4.8% compounded

monthly. Then find the balance after 2 years.

2. Write a compound interest function to model $15,000 invested at a rate of 3% compounded monthly. How long will it take for the amount of investment to reach $20,000?

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Application #2, continued: Compounding Continuously, The number e and the Natural Logarithm There are two numbers for bases that are very important for applications of physics and mathematics.

One is base 10, and the other is base “ .” is an irrational number, like . Because e is so important, there is also a special notation for ,

called “ln.” is a couple of things:

as x gets very close to 0

as h gets very close to 0

(

)

as n gets close to

8 “Compounding Continuously” means that the interest on your (loan, borrowing, investment) is paid out every teeny, tiny, unimaginably, infinitesimally small amount of time. It’s like compounding an “infinite number of times.” .

Example 6: Write a compound interest function to model $1200 invested at a rate of 1% compounded continuously. Then find the balance after 3 years.

Example 7: Write a compound interest function to model $15,000 invested at a rate of 3.05% compounded continuously. How long will it take for the investment to reach $20,000?.

Investments Compounded Continuously A = final amount of $ (balance) P = Principal (Initial) investment e = the irrational number that is approximately 2.718 r = rate expressed as a decimal t= number of years of investment

You don’t need to know any of this this year! It is very important in

finance and in calculus.

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You Try it!

1. Write a compound interest function to model $15,000 invested at a rate of 2.8% compounded

continuously. Then find the balance after 2 years.

2. Write a compound interest function to model $15,000 invested at a rate of 1% compounded continuously. How long will it take for the investment to reach $17,000?.

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Application #3: The pH Scale pH means “power of hydrogen.” It is the concentration of [H+] ions in a substance. For example, HCL, a very strong acid, has a concentration of approximately 3.9 ×10−3. The pH of HCL is about 3. Water, to which all other liquid substances is compared, is completely neutral, with a concentration of 1×10−7 , has a pH of 7.

Example 8: Determine the pH of a substance that has a concentration of H+ of 2.34 x 10-8

Example 9: Determine the concentration of H+ for a substance that has a pH of 12.

You Try it!

1. 2.

where [H+] is the concentration of hydrogen ions

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SUMMARY

Exit Ticket

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Day 9 – Graphs of Logarithmic Functions

Warm - Up:

Using the table below: a) Complete the table of values for y= 2x

b) sketch the graph of y= 2x

x y

-2

-1

0

1

2

2) Recall:

How do we find the inverse of a function?

Find the inverse algebraically.

3) Graph the inverse of the function y = 2x.

Properties of

Properties of

Domain:

Domain:

Range:

Range:

Asymptote:

Asymptote:

x-intercept:

x-intercept:

y-intercept:

y-intercept:

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58

59

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Homework

1. Sketch below the graph of . Then, state the domain and range of the graph. Write the equation of

the asymptote.

2. The expression is equivalent to

1)

2) 2

3)

4)

3. Which expression is not equivalent to ?

1)

2)

3)

4)

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1)

2)

3)

4)


1)

3)

2)

4)

6. Given: and

Express in terms of x and y:

7. Find the value of correct to four decimal places.

8. If , then the solution set for x is

1)

2)

3)

4)

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9. Drew’s parents invested $1,500 in an account such that the value of the investment doubles every seven

years. The value of the investment, V, is determined by the equation , where t represents the

number of years since the money was deposited. How many years, to the nearest tenth of a year, will it take

the value of the investment to reach $1,000,000?

10. Solve for x:

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Algebra 2 and Trigonometry - White Plains Middle School · Algebra 2 and Trigonometry Chapter 8:...

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