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

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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

Page 1 of 25

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 =

Page 2 of 25

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

popu

latio

n (m

illio

ns)

populations5

2376

15281

Page 3 of 25

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

Page 4 of 25

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 4 xy 2) 3.05 .87 xy 3) 4 13 5

x

y

4) 1 32

xy

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 0.5 xy 2) 0.25 xy 3) 15

x

f x

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

12

x

y

1 32

x

y

1 52

x

y

U8D2: Exponential Decay

12

x

y

Page 5 of 25

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 items 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 valueinitial value

r

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

10000150002000025000

1 2 3 4 5

years since purchase

valu

e ($

)

depreciation

0 1 2 3 4

Page 6 of 25

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

12

x

y

12

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 graph 13

x

y

, and then graph

a) 21

3

x

y

b) 1 33

x

y

c)

U8D3: Properties of Exponential Functions & and ex

11 33

x

y

Page 7 of 25

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.

Page 8 of 25

Lets now compare continuous compound interest to other compounds

Two formulas: Continuously: rtA Pe Other: 1ntrA P

n

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 1 23

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

function to 1 23

x ef x ?

Hint: If you are stuck, try your calculator!

Page 9 of 25

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) 41 1

2 16

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

a) 8log 16 b) 641log32

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!

Page 10 of 25

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:___