Definition:



ALGEBRA 2 X

EXPONENTIAL AND LOGARITHMIC FUNCTIONS

|DAY |TOPIC |ASSIGNMENT |

|1 |7.1 EXPONENTIAL FUNCTIONS:GROWTH AND DECAY (Focus on Decay) |P. 493 # 1-9(odd), 10, 15, 31, 32, 34 |

| |p. 494 #15 in class if time | |

|2 |More on Decay and Compound Interest Formula |P. 493 #4, 8, 11, 12 -14,27a, 27b |

| |Do #6 p. 493 in class | |

|3 |Review Days 1 and 2 |TBD |

|4 |7.3 LOGARITHMIC FUNCTIONS |P. 509 # 1-13, 17-25 |

|5 |MORE 7.3(Graphing) USE GRAPH PAPER |P. 509 # 29-30, 36, 40, 41 |

|6 |Review/Quiz 7.1 to 7.3 | |

|7 |7.4 PROPERTIES OF LOGARITHMS |P. 516 # 1-6, 20-25, 40, 41 |

|8 |MORE 7.4 (#48 p. 517 if time) |P. 516 # 7-12, 15,17, 43,45 |

|9 |7.5 EXPONENTIAL AND LOGARITHMIC EQUATIONS |P. 526 # 1- 7, 9-15(odd);21, 23, 25 |

|10 |MORE 7.5 |P. 526 # 10-14 (even) |

| | |22-28(even), |

|11 |Review |TBA |

|12 |TEST ON DAYS 1-11 | |

|13 |Interest Compounded Yearly, Monthly, Continuously etc… |Worksheet in packet (Day 13) |

|14 |7.6 BASE “e”: NATURAL LOGARITHMS |P. 534 # 2,3,5, 17-20 |

| |e=2.71828… | |

|15 |MORE 7.6 |P. 534 # 22,30 – 35, 41, 43 |

|16 |Review (last 3 days) |p. 495 #22 |

| | |p. 534-535 #11, 21, 39 |

|17 |QUIZ (Days 13-16) | |

The exponential function with base b is defined by

        

Where b > _____, b[pic]_____ and x [pic]

| |x |[pic] |

| |f (x) = y | |

|Let's examine the function | | |

|  [pic] |-2 | |

|Check out the table | | |

|and the graph. | | |

| | | |

| |-1 | |

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| | | |

| | | |

| |0 | |

| | | |

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| |1 | |

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| |2 | |

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| |3 | |

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| | | |

|Most exponential graphs resemble this same shape. | |

|This graph is very, very small on its left side and is extremely close to the x-axis.  | |

|As the graph progresses to the right, it starts to ________________________ and shoots | |

|off the top of the graph very quickly, as seen at the right. | |

|In a straight line, the "rate of change" remains the same across the graph.  In these graphs, the "rate of change" |

|____________________________________________________________________________. |

|  |

|Characteristics: |

|     Such exponential graphs of the form  f (x) = bx  have certain characteristics in common: |

|[pic] |  |•  graph crosses the y-axis at (0,1) |

| | |•  when b > 1, _____________________ |

| | |•  when 0 < b < 1, ________________________ |

| | |•  the domain is_______________________ |

| | |•  the range is ________________________ |

| | |•  graph passes the vertical line test – yes or no What does this tell us? |

| | |•  graph is _________________________ to the x-axis - gets very, very close to the |

| | |x-axis but does not touch it or cross it. |

| | |  |

Graph each of these functions on your calculator and draw a rough sketch of each in the space below.

[pic] [pic]

Can either function ever equal zero?

Do the curves ever touch the x-axis? (This has a fancy name.)

What causes the difference in the 2 graphs?

Formula for increasing (growth) or decreasing (decay) functions:

A(t) = a[pic]

A(t) = final amount

a = initial amount

t = number of time periods

Growth vs Decay? Do the following represent growth or decay functions.

a.) [pic] b.) [pic] c.) [pic] d.) [pic]

Growth

1. A house worth $200,000 will increase in value by 4% each year for the next 3 years. A.) Find the value of the house at the end of each of the 3 years. B.) At the end of 10 years, what is the value if rate of change stays the same?

a = _____ b = _______ t = _______

2. 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.

a = _____ b = _______ t = _______

3. 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?

a = _____ b = _______ t = _______

4. 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?

a = _____ b = _______ t = _______

Closure

For an exponential growth function, how do you find b?

Warm Up – Solve and determine if you are working with growth or decay.

Exponential Decay (Decrease) Note: b = 1- rate (r is now negative)

The National Collegiate Athletic Association (NCAA) holds an annual basketball tournament. The top 64 teams in Division 1 are invited to play each spring. When a team loses, it is out of the tournament.

1. How many teams are left in the tournament after the first round of the games?

2. Complete the table (you may have more rows than you need).

|After Round x |No. of Teams Left in |

| |Tournament (y) |

|0 |64 |

|1 | |

|2 | |

|3 | |

|4 | |

|5 | |

|6 | |

|7 | |

|8 | |

3. How many rounds does the winner of the tournament play?

4. Does the graph represent a linear function? How do you know?

5. How does the number of teams left in each round compare to the number of teams in the previous round?

An exponential function can be used to model decay when 0 < b < 1. We call this the decay factor when b < _____. In other words b is a fraction.

If b > 1, we have EXPONENTIAL _____________. If b < 1, we have EXPONENTIAL________.

Word Problems

Decay: (AKA Depreciation)

1. A car worth $30,000 will decrease in value by 20% for the next 3 years. A.) Find the value of the car at the end of each of the 3 years. B.) What is the value after 9 years?

a = _____ b = _______ t = _______

2. The value of a house in 2001 was $175,000 and increases at 8% each year. What is its value in 2009?

a = _____ b = _______ t = _______

2b.) How long will it take for the value to be $320,000? What equation do we have to solve? (we need to use our graphing calculators here)

xmin = -1, xmax = 10, ymin = 0, ymax = 350000, yscl = 1000

3. A motorcycle bought for $10,000 depreciates at 15% each year. Write and graph a function to represent this. What is the value after 5 years?

3b.) When will the value of the motorcycle be $2000? Graph it on your TI and use TRACE to estimate.

How many times would you have to double $1 before you had $512? You could solve this problem if you could solve 2x = 8 by using an inverse operation that undoes raising a base to an exponent equation to model this situation. This operation is called finding the logarithm. A logarithm is the exponent to which a specified base is raised to obtain a given value.

[pic]

|Exponential Equation |Logarithmic Form |

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|[pic] | |

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|[pic] | |

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|[pic] | |

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|[pic] | |

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|[pic] | |

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|[pic] | |

|Logarithmic Form |Exponential Form |

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|[pic] | |

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|[pic] | |

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|[pic] | |

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|[pic] | |

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|[pic] | |

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|[pic] | |

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|[pic] | |

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|[pic] | |

A logarithm with base ______ is called a common logarithm. If no base is written for a logarithm, the base is assumed to be _____.

For example, log 100 = log10100.

Try These by using mental math (convert to exponential form first)

1. log10 2. log 100 3. log 1000

4. log 10,000 5. log 1 6. log [pic]

[pic]

If time, begin tonight’s homework p. 509

Because logarithms are the inverses of exponents, the inverse of an exponential function, such as y = 2x, is a logarithmic function, such as y = log2x.

| |Definition: |

| |The logarithmic function is the function [pic], where b is any number such that [pic]. |

| |                  [pic]is equivalent to [pic] |

| | |

| | |

| |x |[pic] |

|Let's examine the function |f (x) = y | |

|  [pic] | | |

|Rewrite it as an exponential | | |

| |-2 | |

|_______________ | | |

| | | |

| |-1 | |

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| |0 | |

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| |1 | |

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| |2 | |

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| |3 | |

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|Most logarithmic graphs resemble this same shape. | |

|Asymptote_________________ | |

|Domain________________ | |

|Range_________________ | |

|In a straight line, the "rate of change" remains the same across the graph.  In these graphs, |

|the "rate of change" VARIES depending where you are on the graph. |

|  |

Steps to Graphing a logarithm

1. Rewrite the log as an exponential

2. Create a table of values – PLUG IN FOR Y FIRST: -2, -1, 0, 1, 2 is a good start

3. Solve for x and graph.

|X |Y |

| |-2 |

| |-1 |

| |0 |

| |1 |

| |2 |

1. [pic] Rewrite: [pic] Table

Domain:

Range:

Asymptote:

2. [pic] 3. [pic]

Rewrite Rewrite

Domain:

Range:

Asymptote:

4. [pic] 3. [pic]

Rewrite Rewrite

Domain:

Range:

Asymptote:

Quiz Tomorrow on 7.1 to 7.3 – Mini Review

1. A new truck sells for $30,000 and depreciates 11.5% per year.

a. Write the exponential function for the value of the truck after x years.

Formula: [pic]

b. How much is the truck worth after 6 years?

Today we will focus on 3 properties.

I. Product Property of Logarithms

[pic]

Note: One side is written as a single logarithm and the other is expanded.

Example: [pic]

Mental Math Example

(Do this one in your head to show that the property works.)

[pic]

Example: (Your answer is not always numeric so you can’t always simplify.)

[pic]

You try

a.) [pic] b.) [pic] c.) [pic]

II. Quotient Property of Logarithms

[pic]

Example: [pic]

Example: [pic]

You try

a.) [pic] b.) [pic] c.) [pic]

Mixing the Product and Quotient Properties

a.) [pic] b.) [pic]

c.) [pic]

III. Power Property of Logarithms

What is [pic]?______ What is [pic]? ______

Turns out [pic]= [pic].

Power Property: [pic]

Examples (Express as a product and simplify if you can.)

[pic] [pic] [pic]

Practice…

[pic]

Inverse Properties

Rewrite the following in exponential form: [pic]

Rewrite the following in logarithmic form:[pic] (this one hurts my head)

Examples

[pic] [pic]

Logarithms and Calculators

What if you wanted to know [pic].

Our calculators only do base 10 and another base that we’ll learn later.

We could guess. We know [pic]

Let’s see how close we can get to the exact answer by estimating.

If we want to be precise, we use:

[pic]

How it works…

[pic]

Evaluate the following using the Change of Base Formula

(we’ll use base 10 the default)

a.) [pic] b.) [pic] c.) [pic]

Let’s Practice

[pic]

Geology Application – Richter Scale

The tsunami that devastated parts of Asia in December 2004 was spawned by an earthquake with magnitude 9.3. Find the energy released (E).

[pic]

[pic]

[pic]

Finish this – solve for E

Try This

In 1964, loooooooong before your teachers were born, an earthquake centered at Prince William Sound, Alaska, registered a magnitude of 9.2 on the Richter Scale. Find the energy released (E) by the earthquake. USE THE FORMULA

Closure

1.) What is the base of log 17?

2.) Rewrite [pic] as an exponential and solve?

3.) The answer to a log is an ________________.

An exponential equation is an equation containing one or more expressions that have a variable as an exponent.

Method 1 for Solving

• Try writing them so that the bases are all the same.

Method 2 for Solving

• Take the log of both sides

Example – Solve for x and CHECK

[pic]

CHECK

You Try – Solve and Check

a.) [pic] CHECK b.) [pic] CHECK

Method 2 for Solving

• Take the log of both sides

Example

[pic] (5 and 20 can’t be expressed using the same base)

[pic] (take the log of both sides-

what’s our base?)

CHECK – use a calculator

You Try – SOLVE AND CHECK

a.) [pic] b.) [pic] c.) [pic]

Solving Logarithmic Equations

These are all a bit different so we’ll do them together.

Example 1: [pic] CHECK

Example 2: [pic] CHECK

Example 3: [pic] CHECK

Example 4: [pic] CHECK

[pic]

Use this worksheet as a class or partner activity.

[pic]

[pic]

[pic]

The individual sections can be found on page 521.

[pic]

[pic]

Graph [pic] Graph [pic]

(see notes from Day 5 on graphing)

[pic]

Compound Interest

Compounding interest means to give a portion of a total amount spread out over a long period of time.

▪ For example, 12% interest compounded monthly for a year would apply 1% to the principal amount that exists each month.

▪ You receive interest on the interest money you have previously earned.

[pic]

A = final amount; P = principal; r = rate (decimal form); t = time in years

n = # of times interest is compounded per year

Calculate the balances for a deposit of $1000 at 8% at various compounding periods, as shown in the table below. (use your calculator)

You must use ( ) correctly or order of operations

|Number of yearly compoundings=n |Balance in dollars, a |

|Annually, n=1 |[pic]=$1080.00 |

|Semiannually, n=2 |A= |

| | |

|Quarterly, n=4 |A= |

| | |

|Monthly, n=12 |A= |

| | |

|Daily, n=365 |A= |

| | |

As n increases to be really BIG (like compounding every second—what does that mean?

Compound interest for multiple years then will be found through the following formulas, where t = the # of years of compounding:

1. N compoundings per year: [pic]

2. Continuous compounding [pic]

Determine the balance A for p dollars invested in each of the descriptions below.

[pic]

Determine the amount of time, in years that it would take for a $15,000 investment to become worth $100,000 in each of the following situations.

We’ll use the 1 and 2 as examples. Write out the formula. What do we know?

1. 5% interest compounded continuously

2. 12% interest compunded monthly

3. 12% interest compounded continuously________________

4. 7.5% interest compouned quarterly_______________

Ultimately, which is more important to your long term investment, the rate of return, or the type of compounding?

Example on doubling/tripling your money.

How long would it take to triple an investment of $5,000 at a rate of 5% compounded continuously? Which formula would we use.

Homework: Do these problems on a separate sheet, neatly with formulas, work and setup.

Determine the amount of money you will end up with in each of the following scenarios for compounding interest.

1. invest $1200 at 4% compounded quarterly for 20 years

2. invest $2200 at 10% compounded quarterly for 10 years

3. invest $500 at 8% compounded continuously for 20 years

4. invest $1000 at 5.5% compounded monthly for 15 years

5. invest $10,000 at 7% compounded continuously for 30 years

Determine how long it will take to increase your investment to the specified amount in each of the following scenarios.

6. investing $1000 at 6% compounded monthly, to grow to $2500

7. investing $1200 at 3% compounded continuously, to grow to $2500

8. investing $500 at 12% compounded quarterly, to grow to $2500

9. investing $1000 at 4% compounded daily, to grow to $2500

10. investing $15,000 at 5% compounded continuously, to grow to $25,000

11. How long will it take to double any investment if you receive 4% interest compounded continuously?

12. What if you wish to triple your amount of money in this situation?

[pic]

|x |y |

|-1 | |

|0 | |

|1 | |

|10 | |

e is an irrational number like pi. e is a constant – find it out on your TI-##

e = _______________

Graph [pic]

Now graph the line y = e. Where would that be?

Conclusion:

Graphing the natural log.

It looks like any other exponential.

|x |y |

|-2 | |

|-1 | |

|0 | |

|1 | |

|2 | |

|3 | |

|x |y |

|-2 | |

|-1 | |

|0 | |

|1 | |

|2 | |

|3 | |

[pic] [pic]

Now that you know what [pic] looks like let’s try some translations.

Because e is not an integer a table of values is helpful.

1. [pic] 2. [pic]

|x |y |

|-2 | |

|-1 | |

|0 | |

|1 | |

|2 | |

|3 | |

|x |y |

|-2 | |

|-1 | |

|0 | |

|1 | |

|2 | |

|3 | |

|x |y |

|-2 | |

|-1 | |

|0 | |

|1 | |

|2 | |

|3 | |

3. [pic] 4. [pic]

|x |y |

|-2 | |

|-1 | |

|0 | |

|1 | |

|2 | |

|3 | |

Conclusions

1. Describe the difference between [pic]and[pic].

2. Describe the difference between [pic]and[pic].

3. Describe the difference between [pic]and[pic].

Natural Logarithms

A logarithm with a base of e is called a natural logarithm and is abbreviated as “ln” (rather than as loge).

The natural logarithmic function f(x) = ln x is the inverse of the natural exponential function f(x) = ex.

Graph the following on your TI

[pic] [pic] [pic]

How do we know graphically that they are inverses?

Simplifying Expressions

Review might help here. Write [pic] as an exponential.

ln represents the natural log with base e (it’s implied).

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic]

[pic]

Equations (Use properties of logs but for natural logs instead of base 10)

Product:[pic] Quotient: [pic]

Power Rule:(look this up in your packet – try page 14ish)

Solve

1. [pic] 2. ln 7 – ln x = 5 3. ln 6 + ln x = 1

Half-Life Application – What is it?

The term half-life is defined as the time it takes for one-half of the atoms of a radioactive material to disintegrate. Half-lives for various radioisotopes can range from a few microseconds to billions of years.

Determine how long it will take for 650 mg of a sample of chromium-51 which has a half-life of about 28 days to decay to 200 mg.

[pic]

1. Interest and the number e

Example: A Savings account earns 3.75% interest annually. The account has an initial value of $2000.

How much is in the account after 18 years?

A. Interested calculated annually (once per year)

[pic] P = Principal (initial amount); x = time in years

B. Interest calculated Quarterly

(when the interest is compounded more than once a year the formula gets tricky)

C. Interest calculated Monthly

D. Interest calculated Continuously (e)

Unit 7 Quiz 2 Review (use page 553 7-5, 7-6 or stations)70

-----------------------

Day 1: Exponential Functions, Growth and Decay

[pic]

[pic]

b = growth/decay factor

growth: b = 1 + r

decay: b = 1 – r (we’ll get to this one later)

Day 2: Exponential Functions, Growth and Decay cont…

Day 3: Review Days 1 and 2 (and wrap up loose ends)

Day 4: Logarithmic Functions (7.3)

Reading Math

Read logb a= x, as “the log base b of a is x.” Notice that the log is the exponent.

Day 5: Graphing Logarithmic Functions (7.3)

[pic]

[pic]

[pic]

Domain:

Range:

Asymptote:

[pic]

[pic]

Domain:

Range:

Asymptote:

Day 7: Properties Logarithmic Functions (7.4)

Helpful Hint

[pic]

Day 8: Properties Logarithmic Functions (7.4 continued)

The Richter magnitude of an earthquake, M, is related to the energy released in ergs E given by the formula.

M = Magnitude; E=energy released in ergs

Let M = 9.3 for the magnitude of the Asian 2004 earthquake.

Use the quotient property to write this as 2 logs expanded. Solve for E (not e).

11.8

13.95 = log

100

E

æð

çð

èð

öð

÷ð

øð

Day 9: Exponential and Logarithmic Equations (7.5)

How can we represent the numbers 9 and 27 with the same base?

Reminder

When we sol⎛

⎜

⎝

⎞

⎟

⎠

Day 9: Exponential and Logarithmic Equations (7.5)

How can we represent the numbers 9 and 27 with the same base?

Reminder

When we solve equations we use inverses even simple equations like:

3x = 15

(division is the inverse of multiplication)

Day 10: Exponential and Logarithmic Equations (7.5 cont….)

Day 11: Review – BIG TEST 7.1, 7.3 to 7.5

[pic]

[pic]

Day 13: Compounding Interest

Day 14: The Natural Base, e (7.6)

Day 14: The Natural Base, e (7.6 cont…)

Day 16: Review

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