Unit 8: Exponential & Logarithmic · PDF fileDAY TOPIC 1 Exponential Growth 2 Exponential...

DAY TOPIC 1 Exponential Growth

2 Exponential Decay

3 8.2 Properties of Exponential

Functions; Continuous Compound Interest ( xe )

4 8.3 Logarithmic Functions; Converting between log and exp.

5 8.3 Logarithmic Functions;

Inverses; Graphs; Domain and

6 8.4 Properties of Logarithms

7 8 8.5 Exponential and Logarithmic

Equations

9 8.5 Solving Logarithmic Equations

10 Applications of Logarithms 11 8.6 Natural Logs 12

Applications of Natural Logs

13 Review

Date _________ Period_________

Unit 8: Exponential & Logarithmic Functions

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Warmup: Fill in the table below to graph the function 2xy . Then, plot your points and graph!

Check your graph using your graphing calculator. The general form of an _________________________ function (get it: exponential) is xy ab .

How the b value affects the exponential function ...b 0b 0 1b 1b 1b

Example Equation

Graph Looks Like

Name of Function

Additional

Notes

Today we will focus on exponential _________________ & tomorrow will be ________________. The b value, when it is greater than 1, is known as the ______________ __________________. An exponential function can model population growth. If you know the rate of increase r, you can find the growth factor by using the equation

U8D1: Exponential Growth

x 2x y

-3

-2

-1

0

1

2

3

b =

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Example: Refer to the chart at right. In 2000, the annual rate of increase in the U.S. population was about 1.24%. a) Find the growth factor of population. b) Suppose the rate of increase continues to be 1.24%. Write a function to model the population growth. c) Use your model from b) to predict the U.S. populations in 2015 to the nearest million. d) Explain why the model and your prediction may not be valid for 2015 e) Suppose the rate of population increase changes to 1.4%. Write a function to model population growth

and use it to predict the 2015 population to the nearest million Graph each function by creating a table of values. Check on your calculator. #1) 1.5xy #2) 9 3xy

Growth factor = ____________ Growth factor = ____________ Calculator Check Calculator Check

U.S Population 1800-2000

0

100

200

300

1800 1850 1900 1950 2000

year

po

pu

lati

on

(m

illio

ns)

populations

523

76

15

281

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Directions: Write an exponential function xy ab for a graph that includes the given points

Together On Your Own 4,8 , 6,32

2,18 , 5,60.75

Directions: Write a function to model internet usage in the United States. About 84 million homes used in the Internet in 2000, but usage grew about 34% each year from to 2005. Also, use your model to predict the number of homes used the internet in 2005. Wrap up: Describe what happens to the exponential graph xy ab when b fits the following criteria: a) Less than 0 b) Between 0 and 1 c) Equal to 1 d) Greater than 1 Extension: #11 from homework Write the exponential equation thru the points 2,122.5 , 3,857.5

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Today we are going to talk about exponential _________________ This means that the b value is _________________________________ Directions: Without graphing, determine if each equation represents exponential growth or decay.

1) 200 4x

y 2) 3.05 .87x

y 3) 4 1

3 5

x

y

4) 13

2x

y

growth / decay growth / decay growth / decay growth / decay Question: What happens to the graphs as x increases? In fact, the graph approaches those numbers, but never reaches them. This is known as an ___________________________.

Note: this is the beginning of a calculus concept 1

______

Directions: Graph each function and then identify the horizontal asymptote.

1) 2 0.5x

y 2) 0.25x

y 3) 1

5

x

f x

H.A. ____________ H.A. ____________ H.A. ____________

1

2

x

y

13

2

x

y

15

2

x

y

U8D2: Exponential Decay

1

2

x

y

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Additional Note: Just like a growth factor, you need to be able to identify a decay factor. Again, use the formula 1b r , except here r must be ______________________ (as well as a decimal). Depreciation is the decline in an item’s value resulting from age or wear. When an item loses about the same percent of its value each year, you can use an exponential function to model the depreciation. Example: The exponential decay graph shows the expected depreciation for a car over four years. Estimate the value of the car after 6 years. To find r, use the formula:

final value initial value

initial valuer

Write an exponential function to model each situation. Find each amount after the time specified. Be careful, some are growth and some are decay. 1. A tree 3 ft. tall grows 8% each year. How tall will the tree be at the end of 14 years? Round the answer to the nearest hundredth. 2. The price of a new home is $126,000. The value of the home appreciates 2% each year. How much will the home be worth in 10 years? 3. A motorcycle purchased for $9,000 today will be worth 6% less each year. For what can you expect to sell the motorcycle at then end of 5 years? Closure: How do you find the decay factor?

Expected decrease in value

05000

10000

150002000025000

1 2 3 4 5

years since purchase

valu

e ($

)

depreciation

0 1 2 3 4

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Warmup: Use your calculator to graph each of the functions below. Next, analyze the equation and make some generalizations about how they affect the graph.

2xy

2xy

1

2

x

y

1

2

x

y

Describe what you notice: When 0a , the graph of xy ab is a reflection of | | xy a b over the x-axis. Graph 4 ( 2 ) xy

If you know the graph of xy ab (ex: 2xy ), then… the graph x hy ab k , where h moves __________________

and k move ___________________.

Example: First graph1

3

x

y

, and then graph…

a) 2

1

3

x

y

b) 1

33

x

y

c)

U8D3: Properties of Exponential Functions & and ex

11

33

x

y

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Just like , ”e” is an irrational number approximately equal to 2.71828 Exponential functions with a base of e are useful for describing continuous growth or decay. Your graphing calculator has a key for xe Graph xy e and then evaluate the following (to 4 dec. places). a) 2e b) 4e c) 3e NOTE: e is a ____________________, not a _____________________. e is used to find interest when it is ________________________ compounded. The formula is Examples. a) Suppose you invest $1050 at an annual interest rate of 5.5% compounded continuously. How much will you have in the account after 5 years?

b). Suppose you invest $1300 at an annual interest rate of 4.3% compounded continuously. Find the amount you will have in the account after three years.

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Let’s now compare continuous compound interest to other compounds…

Two formulas: Continuously: rtA Pe Other: 1

ntr

A Pn

Key: ______________r , ______________t , ______________n Suppose you invest $2000 at an annual interest rate of 4.5%. How much will you have after 5 years?

Quarterly:

Monthly:

Annually:

Daily:

Continuously: Which method gives you the most? By how much?

Wrap up: If you graphed the function 12

3xf x , how would that graph be affected if you changed the

function to 12

3x ef x ?

Hint: If you are stuck, try your calculator!

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The logarithm is perhaps the single, most useful arithmetic concept in all the sciences; and an understanding of them is essential to an understanding of many scientific ideas. Logarithms may be defined and introduced in several different ways. A logarithmic function is the ______________________ of an exponential function. What does that mean?! (more on this tomorrow)

Logarithm Definition: Log to the base b of a positive number y is defined as Examples: Write each equation in logarithmic form

a) 35 125 b) 2144 12 c) 4

1 1

2 16

Evaluating Logarithms: Write in exponential form, find a common base, set the exponents =, and solve.

a) 8log 16 b) 64

1log

32 c) 9log 27 d) log100

This is known as the ______________________ log (base _______)

Just like 16 implies to take a square root (as opposed to a cube root, etc), log implies base ______! Activity!

U8D4: Logarithmic Functions: Converting between log and exp.

If xy b , then logb y x

FOR LOGS!

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10xy logy x

Asymptote:

Domain:

Range:

Notice, 10xy and logy x are __________________ because they are reflected over the line _________. What are other ways to determine inverses?

Practice Graphing Logs Domain:_______ Range: ________ Example: Graph 3logy x Step 1: Write in Exponential Form Step 2: Make a Table Step 3: Pick values for y, and solve for x (this is backwards of what you’re used to) Step 4: Graph the points & connect Step 5: Check that your graph is reasonable! The book also shows a way to graph using inverses… so let’s look at finding inverses of logs! 1) 0.5logy x 2) 2log 2y x 3) 2

5logy x

U8D5: Logarithmic Functions: Inverses & Domain/Range

3yx x y

Inverse?

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Now that we know how to graph logs, we can use ____________________ to graph other logs!

2log ( 3) 4y x and state the domain and range.

2logy x

Asymptote: _________ Domain: ___________ Range: ___________

logby x is a parent graph, then logby x h k is a translation h units horizontally and k vertically.

Try on your own: Graph 3log 3y x

Asymptote: Domain: ______________ Range: _______________ Wrap up question: What have you notice about the domain of all of our log functions? Let’s generalize: Extension: HW START! (Directions: Find the inverse) 63. 4logy x 65. 10logy x 67. log 1y x 69. log 2y x

x y

2log ( 3) 4y x x y

Translate!

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Complete the table below using your calculator. Round each value to the nearest thousandth.

x 1 2 3 4 5 6 7 8 9 10 15 20 log x

Directions: Complete each pair of statements below by using the information in the table you completed. a. log3 log5 _________________ log(3 5) log(15) __________________ b. 2 log3 _______________________ 2log(3 ) ______________________

c. log9 log3 __________________ 9

log3

= __________________

d. log10 log 2 _______________ log(10 2) log 20 ___________________

e. log8 log 2 ___________________ 4

log2

= __________________

f. log 2 log5 _________________ log(2 5) log10 ____________________ Directions: Find patterns in the above problems to complete the statements below…

Note: M, N, and b are positive #’s, and 1b . Check with Mr. Prosser to ensure that you are correct Then start your flash cards!

U8D6: Properties of Logarithms

Complete the statement: log logM N ________________________

Complete the statement: log logM N ________________________

Complete the statement: logx M ________________________

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Practice with the Properties! State the property or properties used to rewrite each expression

1. 2 2 2log 8 log 4 log 2 2. 3log 3log logb b bx y x y

3. 5 5 5log 2 log 6 log 12

Write each logarithmic expression as a single logarithm

4. 3 3log 20 log 4 5. 2 23log logx y

6. 3log 2 log 4 log16

7. Can you express 2 63log 9 log 9 as a single logarithm? Why/why not?

Expand each logarithm

8. 5logx

y 9. 4log 3r 10. 2log 7b

11.

2

log3

y

12. 3 4

7log a b

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More Practice! There is a saying, “practice makes perfect.” But it is untrue: only perfect practice makes perfect. So please practice perfectly.

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Warm Up: Evaluate each of the following…

1. 3 3

1 1log 2 log

4 4x 2. log1 log100

Write as a single logarithm

3. 2 2log 4logx y 4. 1 1

2log3 log 4 log92 2

An equation of the formcxb a is known as an exponential equation.

Solve by taking the log of both sides…

1. 37 20x 2. 3 4x

3. 26 21x 4.

25 16x To evaluate a logarithm with any base you can use the CHANGE OF BASE FORMULA

loglog

logc

bc

MM

b

Examples:

1. Use the Change of Base formula to evaluate 3log 15. Then convert 3log 15 to a logarithm of base 2.

2. Evaluate 5log 400 and convert it to a logarithm in base 8.

U8D8: Exponential and Logarithmic Equations

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3. Evaluate 6log 12 . Then convert 6log 12 to a logarithm of base 3.

4. Use the change of base formula to solve the exponential equation 32 172x

5. Use the change of base formula to solve the exponential equation 57 3000x

6. Solve 26 1500x using your graphing calculator.

7. Solve 34 1100x with your calculator. Wrap up: Name the 4 log properties from your flash cards

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1. Directions for solving logarithmic equations.

1) Write as a single log (when necessary) 2) Convert to exponential form (do not forget base-jump!) 3) Solve..

2. Solve the logarithmic equation

a. 5)13log( x b. 1)27log( x c. 4)22log( x 3. Use properties of logarithms to solve the equation

a. 23loglog2 x b. 23log6log x c. 52loglog3 x Solve each equation. Round answers to the nearest

U8D9: Solving Logarithmic Equations

4 5 6

87 9

11.10. 12.

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U8D10: Applications of Logarithms

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Remember e? We use e ( approximately 2.71828) to graph some exponential equations of the form xy e Just as xy b has an inverse, xy e has an inverse called a natural log. Natural logs are commonly written as ln x .

The properties of common logs apply to natural logs as well Directions: Simplify. Write each expression as a single logarithm.

1. 3ln 6 ln 8 2. 5ln 2 ln 4 3. 3ln lnx y 4. 1 1

ln 3 ln4 4

x

Solve each equation

5. 2ln(3 5) 4x 6. ln 0.1x 7. ln(3 9) 21x 8. 2

ln 123

x

Use natural logarithms to solve each equation.

9. 27 2.5 20xe 10. 1 30xe 11. 2

5 7.2 9.1x

e 12. An initial investment of $100 is now valued at $149.18. The interest rate is 8%, compounded continuously. How long has the money been invested?

U8D11: Natural Logs

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Mixed Review: Try the following problems as practice for your upcoming test Graph each logarithm (no calculator) 1. 6logy x 2. log( 2)y x

Expand each logarithm

3. 3

5log

s

r 4. 2

6log (3 )xy 5. 6log 4 x

Solve each equation 6. 7 2 1x 7. log5 2x 8. 3log 9x 9. Explain how to use the Change of Base formula to rewrite 2log 10 as a logarithmic expression with base 3.

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Part I: Directions: You must complete the odd problems in this section.

U8D12: Applications of Natural Logs

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1. Consider the function: 2log 1y x

a. Sketch the graph by using a table. b. Identify the asymptote (write its equation!!) c. Find the inverse of the function. d. On the same axis, sketch the inverse.

Be sure to label which is which! 2. Evaluate each logarithm.

a. 3

1log

27 b. 4log 6 c. 5 82 log 10 log 16

3. Write as a single logarithm.

a. 2

3log 2log log3

x y z b. 1

4(ln ln ) 3ln2

xyx y

z

4. Use the properties of logarithms to expand each logarithm.

a. 3

2log

xy

z b. 53ln 2x

5. Solve for x in each equation. Be sure to check your solution!

a. 2log 3 5 0x b. 3log 2x c. 1 520 3 x

U8D13: Review

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d. 3115

4xe e. 4 4log 1 2 log 3 2x x

f. 1 42 3x x g. ln 2 ln 1x x

h. ln( 2) 3xe i. 1

log 23x

Word Problems – You must know the formulas for interest – continuous and specified “n”, half-life and growth/decay problems. Any other formulas needed would be given to you. 6. You invest $750 in an account that earns 6% interest. How long will it take the money to mature (in years) to $2000 if compounded, a. quarterly b. monthly c. continuously 7. The bacteria count of a certain sample is 50 at the beginning of an experiment. If it is growing at a rate of 13% per hour, what will be the count after 5 hours? How long will it take for the bacteria count to be 200?

8. The half-life of a certain radioactive material is 5 days. An initial amount of the material has a mass of 515 kg. How much is remaining after 15 days? After 72 days?

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logb MNPRODUCT PROPERTY

logb

M

N

QUOTIENT PROPERTY

log xb M

POWER PROPERTY

logb M

CHANGE OF BASE PROPERTY

log logb bM N

log

logc

c

M

b

log logb bM N

logbx M

Unit 8: Exponential & Logarithmic · PDF fileDAY TOPIC 1 Exponential Growth 2 Exponential...

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Transcript of Unit 8: Exponential & Logarithmic · PDF fileDAY TOPIC 1 Exponential Growth 2 Exponential...