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Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 1 x y x y x y x y Chapter 7: Exponential and Logarithmic Functions 7.1 Exploring Exponential Models An exponential function is a function looks like: x y ab 0 a , 0 b , 1 b Example 1: Graph the following exponential functions a) 2 x y b) 1 2 x y c) 23 x y d) 1 3 4 x y

Chapter 7: Exponential and Logarithmic Functions€¦ · Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 2 Exponential Behavior Two types of exponential behavior:

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Page 1: Chapter 7: Exponential and Logarithmic Functions€¦ · Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 2 Exponential Behavior Two types of exponential behavior:

Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 1

x

y

x

y

x

y

x

y

Chapter 7: Exponential and Logarithmic Functions 7.1 Exploring Exponential Models

An exponential function is a function looks like:

xy ab 0a , 0b , 1b

Example 1: Graph the following exponential functions

a) 2xy b) 1

2

x

y

c) 2 3x

y d) 1

34

x

y

Page 2: Chapter 7: Exponential and Logarithmic Functions€¦ · Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 2 Exponential Behavior Two types of exponential behavior:

Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 2 Exponential Behavior Two types of exponential behavior: Exponential Growth: as x increases, y increases Exponential Decay: as x increases, y decreases Exponential graphs are also asymptotic. An Asymptote is a line the graph approaches, but never crosses.

Example 2: Write D if the function represents exponential decay. Write G if it represents exponential growth. What is the y-intercept of the graph?

y-int = ______ y-int = ______ y-int = ______ y-int = ______

Page 3: Chapter 7: Exponential and Logarithmic Functions€¦ · Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 2 Exponential Behavior Two types of exponential behavior:

Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 3 Modeling Exponential Growth/Decay

You can model exponential growth/decay with the formula: ( ) (1 )tA t a r

A(t) = Amount after “t” time periods (the ending value) a = initial amount (the starting value) r = the rate of growth (AS A DECIMAL!!!) positive for growth, negative for decay t = the number of time periods Example 3: Draw a line from the function in Column A to the situation it models in Column B.

Example 4: Suppose you invest $500 in a savings account that pays 3.5% annual interest. How much will be in the account after five years?

Page 4: Chapter 7: Exponential and Logarithmic Functions€¦ · Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 2 Exponential Behavior Two types of exponential behavior:

Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 4 Example 4 continued…. Suppose you invest $500 in a savings account that pays 3.5% annual interest. When will the account contain $650?

Example 5: Suppose you deposit $3000 in a savings account that pays interest at an annual rate of 4%. a) If no money is added or withdrawn from the account, how much will be in the account after 10 years? b) When will the account be worth $5000? Example 6: Write an exponential model for each situation. Find the amount after the specified amount of time. a) A population of birds is currently at 200,000 birds. The population is decreasing at a rate of 3.5% per year. How many birds will there be in 50 years? b) An investment of $75,000 increases at a rate of 12.5% per year. What is the value of the investment after 30 years? c) A new truck that sells for $29,000 depreciates 12% per year. What is the value of the truck after 7 years?

Page 5: Chapter 7: Exponential and Logarithmic Functions€¦ · Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 2 Exponential Behavior Two types of exponential behavior:

Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 5

7.2 Properties of Exponential Functions Example1: Graph 2xy Example 2: Graph 2 3xy

Example 3: Graph 52xy Example 4: Graph 2(2)xy

Each of Ex. 2-4 represent a transformation of the parent function 2xy

We talked about transformations in Lesson 2.6 last semester.

x

y

x

y

x

y

x

y

Page 6: Chapter 7: Exponential and Logarithmic Functions€¦ · Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 2 Exponential Behavior Two types of exponential behavior:

Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 6 Recall from 2.6

Example 5:

How does the graph of 24xy compare to the graph of the

parent function 4xy ?

The graph at the right shows 4xy .

Sketch the graph of 24xy on the same set of axes.

Example 6:

How does the graph of 4 2xy compare to the graph of the

parent function 4xy ?

The graph at the right shows 4xy .

Sketch the graph of 4 2xy on the same set of axes

Page 7: Chapter 7: Exponential and Logarithmic Functions€¦ · Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 2 Exponential Behavior Two types of exponential behavior:

Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 7 Example 7: State the parent function. Describe how the graph of each function is a transformation of the parent function.

a) 1

2 63

x

y

b) 51

34

xy

Parent: Parent: Transformations: Transformations:

c) 6

5 8x

y

d) 2 1x

y

Parent: Parent: Transformations: Transformations: Natural Base Exponentials: Base “e”

2.72e

called the “natural base” or “Euler’s number”

used to describe continuous growth or decay

has the same exponential properties as any other exponential base You can evaluate powers of “e” in your calculator using the “e^x” button. This is usually a shift command of the button called “LN”. Example 8: Estimate to four decimal places.

a) 4e b)

2e c)

3

4e

Page 8: Chapter 7: Exponential and Logarithmic Functions€¦ · Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 2 Exponential Behavior Two types of exponential behavior:

Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 8 In 7.1, we talked about interest compounded annually. For continuously compounded interest, we use the number “e”. Modeling Continuous Exponential Growth/Decay

You can model exponential growth/decay with the formula: ( ) r tA t P e

A(t) = Amount after “t” time periods (the ending value) P = initial amount (the starting value) r = the annual rate of growth (AS A DECIMAL!!!) positive for growth, negative for decay t = time in years Example 9: Suppose you won a contest at the start of 9th grade that deposited $3000 in an account that pays 5% annual interest compounded continuously. About how much will be in the account after you graduate from college in 8 years? Example 10: Find the amount in a continuously compounded account for the given conditions. principal: $600 annual interest rate: 4.5% time: 5 years

Page 9: Chapter 7: Exponential and Logarithmic Functions€¦ · Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 2 Exponential Behavior Two types of exponential behavior:

Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 9

7.3 Logarithmic Functions as Inverses

A LOGARITHM is an EXPONENT!!!

52 = 25 The base is _____ . The exponent is_____.

We say “The log base _____ of _____ is _____.”

We write:

491/2 =7 The base is _____ . The exponent is _____.

We say “The log base _____ of ____ is _____.”

We write:

103=1000 The base is _____. The exponent is _____.

We say “The log base _____ of _____ is _______.”

We write: Example 1: What is the logarithmic form of each equation?

a) 26 36 b)

32 8

3 27

c)

01 3

Example 2: What is the exponential form of each equation?

a) 3log 81 4 b) 16log 8 0.75 c) 2

15 log

32

A common logarithm is a log base 10. You can write 10log logx x , without showing the 10.

Ex. 210 100 Ex. log40 1.6

When 0b and 1b

xb n

if and only if

logb n x

The logarithm is the exponent you must use on base “b” to get the number “n”

Page 10: Chapter 7: Exponential and Logarithmic Functions€¦ · Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 2 Exponential Behavior Two types of exponential behavior:

Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 10 Evaluating Logarithms

What is the value of 5log 125 ?

This question is asking you “What EXPONENT do I need on base 5 to get 125?” Sometimes in problems like this, it is easy to “see” what the answer should be. When it isn’t so easy, follow these steps.

Example 3: Evaluate each logarithm.

a) 4log 64 b) 2log 32 c) 9log 27

d) 8log 32 e) 7

1log

49 f) 125

1log

25

Exponential Property of Equality For 0b and 1b ,

If x yb b

Then x y

Step 1: Write the logarithm. Set equal to x, or some other variable. Step 2: Rewrite the logarithm as an exponential. Step 3: Rewrite both sides of the equation to use the same base. Step 4: Use the property of equality (box upper right) to set exponents equal. Step 5: Solve for your variable.

Page 11: Chapter 7: Exponential and Logarithmic Functions€¦ · Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 2 Exponential Behavior Two types of exponential behavior:

Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 11

x

y

Logarithms and exponential functions are inverses of each other. Example 4:

Graph 2xy .

Then use that graph to draw the graph of

2logy x

Transformations work with logarithmic functions, too.

Example 5: How does the graph of the logarithmic function compare to the graph of the parent function?

a) 2log ( 3) 4y x b) 25logy x c) log( 4) 5y x

(see graph above)

Page 12: Chapter 7: Exponential and Logarithmic Functions€¦ · Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 2 Exponential Behavior Two types of exponential behavior:

Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 12

7.4 Properties of Logarithms A logarithm is an exponent! Therefore, the properties of logarithms corresponds to the properties of exponents.

Example 1: What is each expression written as a single logarithm? Simplify if possible.

a) 3 3log 9 log 24 b) 2 2log 7 log 9 c) 3

4log 16

d) 7 73log 5logx y e) 4log 3logx x f) 2 2log 16 log 8

g) 4 4log 5 log 3x x h) 4 4 4

12log 6 log 9 log 27

3 i) 5

2log 8

Page 13: Chapter 7: Exponential and Logarithmic Functions€¦ · Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 2 Exponential Behavior Two types of exponential behavior:

Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 13 Your calculator can only estimate common logarithms (log base 10). Example 2: Use your calculator to estimate the following to four decimal places.

a) log1000 b) log 45 c) log0.003

To calculate a logarithm that is NOT base 10, you can use the change of base formula. Example 3: Use the change of base formula to evaluate each expression. Round your answer to four decimal places.

a) 2log 32 b) 4log 70 c) 8log 1000

7.5 Exponential and Logarithmic Equations Solving Exponential Equations

With Common Bases:

Example 1: 6 42 2x x Example 2: 327 81x

Exponential Property of Equality For 0b and 1b ,

If x yb b

Then x y

Page 14: Chapter 7: Exponential and Logarithmic Functions€¦ · Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 2 Exponential Behavior Two types of exponential behavior:

Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 14

Not Common Bases:

Example 3: 5 130x

Example 4: 37 24 184x Example 5:

5 42 7x

Logarithmic Equations

One Log in the Equation

Example 6: log(2 4) 3x

Step 1: Isolate the exponential. Step 2: Take the common log of both sides of the equation. Step 3: Use the power property for logarithms to bring down the exponent. Step 4: Solve the equation for x. Step 5: Use your calculator to round your answer to four decimal places

Step 1: Rewrite the logarithm as an exponential. Step 2: Simplify and Solve Step 3: Check your answer.

Page 15: Chapter 7: Exponential and Logarithmic Functions€¦ · Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 2 Exponential Behavior Two types of exponential behavior:

Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 15

Example 7: log 4 3 2x Example 8: log(2 8) 1x

Multiple Logs in an Equation Example 9: 2log log 4 3x

Example 10: log( 3) log 4 2x Example 11: log10 log 2 3x

Step 1: Combine the logarithms using the properties of logarithms. Step 2: Rewrite as an exponential. Step 3: Simplify and Solve. Step 4: Check your answer.

Page 16: Chapter 7: Exponential and Logarithmic Functions€¦ · Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 2 Exponential Behavior Two types of exponential behavior:

Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 16

7.6 Natural Logarithms Recall that the inverse of an exponential is a logarithm. Inverses

The “natural exponential” xe also has an inverse:

Natural logarithm log lne x x Example 1: Rewrite in the inverse format exponential logarithmic or logarithmic exponential

a) 2e x b) 27xe c) ln 3x d) ln 4 x

The properties of logarithms still apply to natural logarithms as well. Example 2: What is each expression written as a single logarithm?

a) ln7 2ln5 b) 3ln 2ln2x x

c) 3ln 2ln ln5x y

2

3

2 and log

3 and log

10 and log

x

x

x

x

x

x

Page 17: Chapter 7: Exponential and Logarithmic Functions€¦ · Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 2 Exponential Behavior Two types of exponential behavior:

Algebra 2B: Chapter 7 Notes Exponential and Logarithmic Functions 17 Example 3: Solve. Round your answer to four decimal places as necessary.

a) ln 2x b) 2ln(3 5) 4x

c) ln2 ln3 2x d) 2 12xe

e) 2 20xe f) 3 5 15xe