Warm-up 7.1

1) What are the zeros of the function : y = x3 – 4x2 + x + 6?

2) What is the discriminant of 2x2 – 5x + 4=0?

3) What is the end behavior of the function:f(x) = 5x4 -3x3 + x – 2?

Did You Know?Q. What do bulletproof vests, fire escapes, windshield wipers, and laser printers all have in common?A. All invented by women.F: One roach can live on a piece of gum for 5 years.

Q. If you were to spell out numbers, how far would you have to go until you would find the letter "A"?A. One thousand

• When a coffee seed is planted, it takes five years to yield it's first consumable fruit.

• The common goldfish is the only animal that can see both infra-red and ultra-violet light.

• Tennessee is bordered by more states than any other. The eight states are Kentucky, Missouri, Arkansas, Mississippi, Alabama, Georgia, North Carolina and Virginia.

• Des Moines has the highest per capita Jello consumption in the U.S

• The Western-most point in the contiguous United States is Cape Alava, Washington.

• There are only three animals with blue tongues, the Black Bear, the Chow Chow dog and the blue-tongued lizard.

Grades, Papers, Folders

Topic:

Exponential and Logarithmic Functions

Key Learning(s):

There are many things in life that are modeled with exponential functions. In order to solve exponential functions, it is helpful to know how to solve their inverses (logarithmic functions).

Unit Essential Question (UEQ):

What real-life situations can you model with exponential and logarithmic functions?

Concept I (7.1):

Exploring Exponential Models

Lesson Essential Question (LEQ):

How are exponential models similar to and different from linear or quadratic models?

Vocabulary:

Decay factor, exponential decay, exponential function, growth factor

QCC’S: #37

Concept II (7.2):

Exponential Functions

Lesson Essential Question (LEQ):

What are the roles of the constants in the formula y = ab (kx)

Vocabulary:

Compound Interest, Half-life

QCC’S: #37

Concept III (7.3):

Logarithmic Functions as Inverses

Lesson Essential Question (LEQ):

How do you convert a problem from exponential to logarithmic form & vice versa?

Vocabulary:

Common logarithm, logarithm

QCC’S: #34

Concept IV (7.4):

Properties of Logarithms

Lesson Essential Question (LEQ):

Why do we need properties of logarithms?

KQ: What are the properties of logs?

Vocabulary:

QCC’S: #33, 37

Concept V (7.5):

Exponential and Logarithmic Functions

Lesson Essential Question (LEQ):

How do you convert from radical to fractional exponent form?

Vocabulary:

Change of base formula

QCC’S: #35, 36, 37

Concept VI (7.6):

Natural Logarithms

Lesson Essential Question (LEQ):

How are natural logarithms related to the number “e”?

Vocabulary:

Natural logarithm

QCC’S: 35, 36, 37

§7.1: Exponential Models

LEQ: How are exponential models similar to and different from linear and quadratic models?

Introduction – Exponential Decay

Exponential Function

Y = abx

“a” is a non-zero constant

“b” is the base (greater than 0 and not equal to 1)

“x” is any real number

Exponential Growth

Y = abx

When b > 1, the function models exponential

growth

“b” is the growth factor

6

5

4

3

2

1

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-1

-2

-3

-4

-5

Exponential Growth

Exponential Decay

y = abx

When “b” is between 0 and 1 (is a positive fraction), the function

models exponential decay.

“b” is the decay factor

6

5

4

3

2

1

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-1

-2

-3

-4

-5

Exponential Decay

Identify each as exponential growth or decay. What is the percent increase/decrease?

1) Y = 1298(1.63)x

2) Y = 2(0.65)x

3) Y = 12(1.7)x

4) Y = 16(0.25)x

1) growth, 63%

2) Decay; 35%

3) Growth, 70%

4) Decay; 75%

Real World

The population of the United States in 1994 was about 260 million, with an average annual rate of increase of about 0.7%.

FYI - Depreciation is the decline in an items value (exponential decay).

What is the growth factor for the U.S. population?

After one year, the population would be 260 + 260(0.7%) million.

= 260 + 260(0.7%)= 260 + 260(0.007)= 260(1 + 0.007)= 260(1.007)

The growth factor is 1.007

Write an equation that models the future growth of the U.S. population.

To write an equation, find “a” and “b”.X = # years after 1994Y = the population (in millions) The population increases exponentially:Y = abx

Y = 260(1.0007)x

Use the equation to predict the population of the U.S. in 2001.

Y = 260(1.0007)x

X is the # of years since 1994, so x = 7

Plug it in…

The population would be about 273 million.

A handy “formula”

• Percent increase/decrease refers to the “slope” of any given line, but typically asks for decrease/increase on a changeable slope

(initial value – final value)Initial value

• Cars are an example of an item that depreciates

exponentially.• Can you think of five other

items that might depreciate exponentially?

• How do you determine whether an equation shows growth or decay?

• How do you calculate the percent increase/decrease of an exponential growth equation?

• How do you determine if a graphs shows exponential decay or growth?

Assignment

Section 7.1p. 308 - 309

#5, 7, 9, 10, 11, 19 – 23

(10 problems total)Homework:

Warm – up 10/22/08

Complete Technology Connection

on pg. 311 in textbook. #1-4

Did you know?• In 1886, Coca-cola was first served at a pharmacy in

Atlanta, Georgia for only five cents a glass. A pharmacist named John Pemberton created the formula for Coca-cola

• Flamingos are able to fly at a speed of approximately 55 kilometers an hour. In one night they can travel about 600 km

• Men are able to read fine print better than women can

• On average, 150 couples get married in Las Vegas each day

• Spiders usually have eight eyes, but still they cannot see that well

• In humans, the epidermal layer of skin, which consists of many layers of skin regenerates every 27 days

• Camel's milk does not curdle. • The ball on top of a flagpole is called the truck. • People generally read 25% slower from a computer

screen compared to paper • Certain female species of spiders such as the Australian

crab spider, sacrifice their bodies as a food source for their offspring

• One grape vine produce can produce about 20 to 30 glasses of wine.

• The Hubble telescope is so powerful that it is like pointing a beam of light at a dime that is two hundred miles away.

• On average people fear spiders more than they do death

• Every day, over five billion gallons of water are flushed down toilets in the United States

• In one trip, a honey bee visits about 75 flowers • Jupiter is the fastest rotating planet, which can

complete one revolution in less than ten hours • A chicken loses its feathers when it becomes stressed

Review?

• How do you determine whether an equation shows growth or decay?

• How do you calculate the percent increase/decrease of an exponential growth equation?

• How do you determine if a graphs shows exponential decay or growth?

Assignment

Section 7.1p. 308 - 309

#5, 7, 9, 10, 11, 19 – 23

(10 problems total)

Applications of exponential

Equations…

7.2: Exponential Functions

LEQ: What are the roles of the constants in the formula y = ab (kx) ?

Compound Interest FormulaA = P(1 + r/n)nt

P = principal t = time in years

R = rate of interestN = # of times per year interest is

compounded

Finding Compound Interest# times compounded per year (n)

P(1 + r/n)nt

P = 1000

R = 10%

Value of A

Annually (1) 1000(1+.1/1)1(1) 1100

Semi-Annually(2) 1000(1+.1/2)2(1)

Quarterly (4) 1000(1+.1/4)4(1)

Monthly (12) 1000(1+.1/12)12(1)

Daily (365) 1000(1+.1/365)356(1)

Hourly (8760) 1000(1+.1/8760)8760(1)

By the Minute

(525,600)1000(1+.1/525600)(525600)(1)

Ex.Write an equation to describe the exponential function of the form y = abx with base 2 whose graph passes through the point (3,4).

4 = a23

4 = a(8)½ = aThe equation would be : y = ½ 2 x

You try:

Write an equation to describe the exponential function y = abx with base 4 whose graph passes through the point (2,3).

y = 3/16 4x

Half-LifeHalf life of a radioactive substance is the time it takes for half of the material to decay or decompose (thus, it has a decay factor or 0.5)

Half Life Formulay = abt/h

a = initial amount b = 0.5 (half-life)t = time passed h = half life of item

Suppose a hospital prepares a 100mg supply of technetium-99m which has a half-life of 6 hours.Find the amount after 75 hours.a = 100 (initial amount)b = 0.5 (half life decay factor)

** Half life is every 6 hourst = 75 hoursAmount left = 100(0.5)75/6

Continuously Compounded Interest

Using the compound interest formula, evaluate (1 + 1/n)n for n = 2, 8, 32, 128, and 512.

What do you think would happen as n continues to increase?

Now find the value of e. (ex)

Continuous Interest

As the compound interest formula approaches ∞, it nears the value of e.

Thus, the continuously compounded interest formula is

A = Pert

Compound Interest Example1) Suppose you invest $1050 at an annual

interest rate of 5.5% compounded continuously. Find the amount you will have in the account after 5 years.

2) How long will you need to leave the money in the account to double your initial investment?

3) Suppose you invest $1300 at an annual interest rate of 4.3% compounded continuously. How much will you have after 3 years?

More Examples/

Applications

Summary Questions

Assignment

Section 7.2p.316

#16 – 22 Even31 - 33,35 - 39

Warm-up 7.3Identify each function as exponential growth or decay & identify % change.

1)Y = 15(1.45) x 2) y = 7.3(0.8)x

3)Describe a function that would be modeled by exponential growth.

4)A radioactive element has a half-life of 30 hours. Write the exponential decay function for the 100-mg sample. Use the function to find the amount of the element remaining after 100 hours.

Did you know?• Sharks are immune to cancer • Manicuring the nails has been done by people

for more than 4,000 years • The study of the iris of the eye is called iridology • Back in 1919, the Russian transplant pioneer

Serge Voronoff made headlines by grafting monkey testicles onto human males’.

• In 1946, the New York Yankees became the first baseball team to travel by plane

• By recycling just one glass bottle, the amount of energy that is being saved is enough to light a 100 watt bulb for four hours

• The Mall of America, located in Bloomington, Minnesota is so big that it can hold 24,336 school buses

• If you have three quarters, four dimes, and four pennies, you have$1.19. You also have the largest amount of money in coins without being able to make change for a dollar.

• In the United States, poisoning is the fourth leading cause of death among children

• In 1916, Charlie Chaplin was making $10,000 a week, making him the highest paid actor of his time

• It's possible to lead a cow upstairs...but not downstairs. • People that smoke have 10 times as many wrinkles as a

person that does not smoke • Thomas Edison was afraid of the dark. (Hence, the light

bulb?)

Assignment

10 minutes; compare with a partner;

Stay with partner, you receive “hw grade” if you get question

correct when called.

Section 7.2p.316

#16 – 22 Even31 - 33,35 - 39

Summary Questions

More Examples/

Applications

Assignment

1) Buying your first new car

2) Financial Futures

3) Half-Life Handout

Homework

7.1-7.2 Quiz

Warm-up 3/20/091) A 3000-mg sample of a certain

radioactive element has a half-life of 3 seconds. How much of the sample remains after 1 minute?

2) For the years 1970-1991, the median price of a single – family home in the U.S. can be approximated by the exponential function A = 25690(1.0725)t, where t is in years (1970=0). What is the growth rate of housing prices for this period?

Did you know?• In Colorado, there are about 83,000 dairy cows • Just by recycling one aluminum can, enough energy

would be saved to have a TV run for three hours. • The first telephone call from the White House was

from Rutherford Hayes to Alexander Graham Bell • Turtles can breathe through their butts • A glockenspiel is a musical instrument that is like a

xylophone. It has a series of metal bars and is played with two hammers

• Teenage suicide is the second cause of death in the state of Wisconsin

• Diamonds were first discovered in the riverbeds of the Golconda region of India over 4,000 years ago.

• French artist, Michel Vienkot, uses cow dung as paint when he creates his pictures

• There are 122 pebbles per square inch on a Spalding basketball

• The seventeenth president of the United States, Andrew Johnson did not know how to read until he was 17 years old

• The fastest growing tissue in the human body is hair • A cubic yard of air weighs about 2 pounds at sea level. • Pancakes are served for breakfast, lunch and dinner in

Australia • A lion feeds once every three to four days • A honey bee has four wings • Cheddar cheese is the best selling cheese in the USA • In the movie "The Matrix Reloaded" a 17 minute battle

scene cost over $40 million to produce

Collect

1) Buying your first new car

2) Financial Futures

3) 7.1-7.2 Quiz

Today’s Lesson

Conversions With

ExponentialsAnd

Logarithms

§7.3: Converting Exponentials

LEQ: How do you convert between exponential and logarithmic forms?

Logarithms are the “opposite” of exponents; they “undo” exponentials.

Logarithms have a specific relationship to exponentials…

The relationshipExponential Equation (a = 1):

y = bx

is equivalent to:

Logarithmic Equation:logb(y) = x

“Log-base-b of y equals x.”Re-written so the exponent

is after the equal sign.

Ex. Convert 63 = 216 to the equivalent logarithmic form.

Base is 6, exponent is 3…log6(216) = 3

Ex. Convert log4(1024) = 5 to the equivalent exponential expression.

Base is 4, exponent is 545 = 1024

Exponent in a different form1) log443= 3

43 = 43

2) log552 = x

5x = 52

3) log225 = y

2y = 25

y = 54) logbb2 = z

bz = b2

z = 2

From Logs to Exponents

1) 52 = 25base is 5, exp is 2: log5(25) = 2

2) (1/2)3 = 1/8log(1/2)(1/8) = 33) 24 = 64log2(64) = 4

To evaluate logarithms, you can write them in exponential form…

Evaluating Exponents

Ex. Log816

8x = 16Both can be written with a base of 2:

23x = 24

Now you can set the exponents equal (since the bases are equal)

3x = 4 Solve for xX = 4/3

So, log816 = 4/3

Ex)Evaluate log5125.

5x = 1255x = 53

Since both bases are five, you can set the exponents equal.

X = 3Thus, log5125 = 3

A short cut

To evaluate log216, you can ask yourself

“What power of 2 is equal to 16”. 1) What question would you ask to

evaluate log327? Evaluate it.

2) What question would you ask to evaluate log10100? Evaluate it.

General Forms

1) What is the value of logb1?a) Log21 c) Log31

b) Log41 d) Log51

2) What is the value of logbb?a) log22 c) log44

b) Log55 d) log99

3) Explain why the base b in y = logbx cannot equal 1.

Inverse GraphsLogarithms and exponentials are also

inverses of each other. If you graph y = 10x and y = logx, they

are inverses. Their graphs are reflections of each other over the line

y = x.*log has a general base of 10*

Find the inverse function of each:1) Y = 3x 2) y = 5x 3) y =bx

1) Y = log3x 2) y = log5x 3) y =logbx

•Wb examples

SO

How do you convert between exponential

and logarithmic forms?

Assignment

Section 7.3p.321-323

#2 -12, 16 - 26, 30 - 40, 50 Even

Warm-up 7.4

1) Write 5-2 = 1/25 in logarithmic form.

2) Write log381 = 4 in exponential form.

3) Evaluate log10,000

Interesting Facts

• Spencer Eldon was the name of the naked baby on the cover of Nirvana's album

• All three major 1996 Presidential candidates, Clinton, Dole and Perot, are left-handed.

• The Madagascan Hissing Cockroach is one of the few insects who give birth to live young, rather than laying eggs.

• The book of Esther in the Bible is the only book which does not mention the name of God.

• Sheriff came from Shire Reeve. During early years of feudal rule in England, each shire had a reeve who was the law for that shire. When the term was brought to the United States it was shortened to Sheriff.

• An animal epidemic is called an epizootic. • Dracula is the most filmed story of all time,

Dr. Jekyll and Mr. Hyde is second and Oliver Twist is third.

• The silhouette on the NBA logo is Jerry West.

• The silhouette on the Major League Baseball logo is Harmon Killebrew.

Check HW

Section 7.3p.321-323

#2 -12, 16 - 26, 30 - 40, 50 Even

Go OverQuestions?

Graphing Inverses

Complete math toolboxp.324

#1 - 9 oddExponentials in the Real World

http://www.thefutureschannel.com/dockets/algebra/communication_speed_light/

§7.4: Properties of Logarithms

LEQ: Why do we need properties of logs?

What are the properties of logs?

Complete “work together”p.325 #1 - 7

Properties of Logarithms

1) logbMN = logbM + logbNProduct Property

2) logbM/N = logbM – logbNQuotient Property

3) logbMk= klogbMPower Property

Examples

Write the expression in single log form.1) log320 – log34

log320/4 = log35

2) 3log2x + log2y

log2x3 + log2y= log2x3y

3) 3log2 + log4 – log 16log(23 x 4)/16 = log 32/16 = log2

Examples of expansion

Expand each logarithm.1) log5(x/y)

log5x – log5y

2) log3r4 log3 + logr4 = log3 + 4 log r

3) Can you expand log3(2x + 1)?

NO, the sum can’t be factored.

Estimating Answers

http://www.algebra.com/algebra/homework/logarithm/Properties-of-Logarithms.lesson

Evaluating Expressions

To evaluate an expression, apply rules, then use the logarithmic rules to solve.

Ex. log55 – log5125

log5(5/125)

log5(1/25)

log5(1/52)5x =5-2

x = -2

Summary

Rewrite each expression

1)logx + log y

2)2log3x – log3x

3)log3 + log4

4)Evaluate: log28 – log22

7.1 – 7.4 Quiz

AssignmentSection 7.4P. 328 – 329

#2 – 34, 38 – 68Even

Extra Practice?http://www-math.cudenver.edu/~rbyrne/flash.htmhttp://www.purplemath.com/modules/index.htm

Warm-up 7.1-7.4

Rewrite each expression

1) logx + logy2) 2logx – logx3) log3 + log4Evaluate the

expression4) log28 – log22

1) log(xy)2) logx3) log12

4) 2

Interesting Facts• Murphy's Oil Soap is the chemical

most commonly used to clean elephants.

• Goat's eyes have rectangular pupils. • Walt Disney's autograph bears no

resemblance to the famous Disney logo.

• Other than humans, black lemurs are the only primates that may have blue eyes.

• The United States has never lost a war in which mules were used.

• Artist Constantino Brumidi fell from the done of the U.S. Capitol while painting a mural around the rim. He died four months later.

• Since 1896, the beginning of the modern Olympics, only Greece and Australia have participated in every Games.

• Cathy Rigby is the only woman to pose nude for Sports Illustrated. (August 1972)

• Blueberry Jelly Bellies were created especially for Ronald Reagan.

Check HW

Section 7.4P. 328 – 329

#2 – 68Multiples of 4

§7.5: Exponential & Log Equations

LEQ: How do you convert from radical to fractional exponent form?

An example:The formula F = km2/3 describes the amount of food F, in kilograms, that a

mammal must eat daily. (k is a constant of variation, and m is the

mass of the animal).

1) Substitute these values for F and k into the given formula for a large elephant, F = 145 kg and k = 0.421. Estimate the value of m2/3.

2) Estimate the value of m. (use a calculator to find a value of m for which m2/3 is near your estimate from Q1).

3) How could you solve the equation F = km2/3 for m2/3? What is the result if each side of the equation is then raised to the 3/2 power.

4) Solve the equation to find the typical body mass of a large elephant. How does this result compare with your estimate from Question 2?

5) For a raccoon, F = 2kg and k = 0.431. Find the typical body mass of a raccoon in kilograms.

Radical Equation

To solve equations in the form xa = c, where the variable is raised to a power, you can either use the properties of exponents or use radicals.

Radical Equation – The variable in an equation occurs in a radicand.Ex. 4 + x2/3 = 31

Solve with reciprocal exponents:

4 + x2/3 = 31x2/3 = (31 – 4)

x2/3= 27Multiply both sides by reciprocal of

3/2x(2/3)(3/2) = 273/2

x = 273/2

x =

Try these1) 3y4/3 = 7682) 6√(10y) = 33) √(7x) – 4 = 04) 102x = 1,000,0005) Why is it

important to check the possible solutions?

1) 642) 72.93) 16/74) 35) Squaring both

sides of an equation can lead to extraneous solutions.

Word ProblemEx2. Meteorologists use the formula

D3= 216T2 to describe the size and duration of storms, such as tornadoes, thunderstorms, hurricanes, and cyclones. In the formula, D is the diameter of the storm in miles and T is the duration, or the number of hours the storm lasts. If the diameter of a thunderstorm is 12 mi, about how long would this storm last?

The diameter of the storm is 12 mi.Solve for T.D3= 216T2

(12) 3 = 216T2

1728 = 216T2

8 = T2

T = √8 = 2.828 hours

#10

Suppose a hurricane lasts for 6 days. Estimate the diameter of the storm.

D3= 216T2

D3= 216(6)2

*Remember T is in hours, 6 days = 144 hoursD3 = 216(144)2

D3 = 4478976D = about 164 miles

Exponential Equations

An equation in the form a = bcx, where the exponent includes a variable is called an exponential equation.

Solve these equations by taking the logarithm of both sides.

Solving Logarithm Tips:

• Logarithms should contain the same base (or you have to use a formula to change them)

• You can only plug positive numbers into a logarithm

• If you have two logs in a problem, one on each side of the equal sign and both with a coefficient of one, you can “drop” the logarithms

Solve2log9(√x) – log9(6x – 1) = 0

2log9(√x) = log9(6x – 1)

log9(√x)2= log9(6x – 1)

x= 6x – 11 = 5x1/5 = x

Always plug the answer back in to make sure that it won’t produce any

negatives or zeros in the logarithms.

Solvelogx + log(x – 1) = log(3x + 12)

log(x(x – 1)) = log(3x + 12)x2 – x = 3x + 12x2 – 4x - 12 = 0

(x – 6)(x + 2) = 0X = 6, -2

When you plug them back in, 6 works, but -2 is an extraneous solution.

Solveln10 – ln(7 – x) = lnx

ln10/(7 – x) = lnx10 = x

7 – x10 = x(7 – x)10 = 7x – x2

x2 – 7x + 10 = 0(x – 5)(x – 2) = 0

x = 5, 2Both are solutions.

Ex. 73x = 20log73x = log20Use rules of exponents to get:3xlog7 = log20Divide both sides by 3log7x = log20/3log7x = 0.513

Try these

1) 3x = 4

2) 62x = 21

3) 3x+4 = 101

1) 1.262

2) 0.850

3) 0.201

Sometimes it’s more useful to convert an

equation to exponential form to

work a problem.

More Practice

Solvelogx = 1 – log(x – 3)logx + log(x – 3) = 1

log(x(x – 3) = 1log(x2 – 3x) = 1

x2 – 3x = 101

x2 – 3x – 10 = 0(x – 5)(x + 2) = 0

x = 5, -2-2 would create a negative log,

5 works, so the only solution is 5.

Solve

Log2(x2 – 6x) = 3 + log2(1 – x)

Log2(x2 – 6x) - log2(1 – x) = 3

Log2(x2 – 6x) = 3

(1 – x)

(x2 – 6x) = 23

(1 – x)

(x2 – 6x) = 8 (1 – x)(x2 – 6x) = 8(1 – x)

x2 – 6x = 8 – 8xx2 – 6x +8x = 8x2 +2x - 8 = 0

(x + 4)(x – 2) = 0x = -4, 2

The only solution that works in the problem is -4 (2 is extraneous)

What about other bases?

To evaluate a logarithm with any base, you can use the change of base formula.

*for this formula, the bases cannot = 1.logbM = log10M

log10b

Ex. Log35 = log5/log3

Ex. of solving with change of base

62x = 1500log662x = log61500

2xlog66 = log61500

log66 = 1…so

2x = log1500/log6 (change of base)2x = 4.0816x = 2.0408

Ex5

a) Solve log(3x + 1) = 5log (101) undo each other = 1, so10log(3x + 1) = 105

3x + 1 = 105

3x + 1 = 100,0003x = 99,999x = 33,333

Ex6b) Solve 2logx – log3 = 2

use rules of exponentslog x2 = 2

3101log(x2/3) = 102

x2/3 = 102

x2 = 102(3)x2 = 100(3)x = √300 = ±17.32

7.5 Assignment/HW

Section 7.5p. 334 – 336

#5 – 60(multiples of 5)

Warm-up 10/28/087.5 Lesson Quiz

• On ½ sheet of paper, complete & solve the following 4 problems.

1) x3/2 = 272) 2 + 3x = 82

3) log x + log 4 = 34) log 3 22 = x

Interesting Facts• Will Clark of the Texas Rangers is a direct

descendant of William Clark of Lewis and Clark.

• When ocean tides are at their highest, they are called "spring tides." When they are at their lowest, they are call "neap tides."

• February 1865 is the only month in recorded history not to have a full moon.

• The last NASCAR driver to serve jail time for running moonshine was Buddy Arrington.

• Many Japanese golfers carry "hole-in-one" insurance, because it is traditional in Japan to share one's good luck by sending gifts to all your friends when you get an "ace." The price for what the Japanese term an "albatross" can often reach $10,000.

• The ridges on the sides of coins are called reeding. • Montpelier, Vermont is the only U.S. state capital without a

McDonalds. • The Roman emperor Caligula made his horse a senator. • At latitude 60 degrees south you can sail all the way around the

world. • A Chinese checkerboard has 121 holes. • The hyoid bone, in your throat, is the only bone in the body not

attached to another bone. • Mice, whales, elephants, giraffes and man all have seven neck

vertebrae. • Sunbeams that shine down through the clouds are called

crespucular rays. • Very small clouds that look like they have been broken off of bigger

clouds are called scuds.

Extra Credit PaperDue: Tuesday – Election Day

1) Minimum length = 2 pgs. 2) Size 12 font, double spaced.

3) Will be graded for correctness & content.4) Include a reference page (MLA/APA ok)

5) Guidelines?

Wednesday

• Tomorrow’s Agenda• 9th graders to the auditorium

For making it count presentation.

2nd block, students will be in a classroom with seniors making

presentations.

10th graders will be taking the PSAT*college scholarships are rewarded based

on PSAT scores• Chadwick’s HR to Davis’ Room

• Danforth’s HR to Parker’s Room• McFather’s HR to Hankins Smith’s Room

• Montgomery’s HR in her room• Ortiz’s HR in Sirman’s Room

• 11th graders will the taking the ASVAB in the Café Annex

• 12th graders to their assigned rooms• Phippen’s HR to Bollinger’s Room• Cochran’s HR to Media Center• Ragan’s HR to her room• School’s HR to her room

Questions?

LastSection…

On ln

§7.6: Natural Logarithms

LEQ: How are natural logs related to the letter “e”?

From 7.2, we learned that the number “e” = 2.71828… can be used as a base of exponents.

Just like logarithms, the function ex has an inverse, called the natural logarithmic function.

The natural log function is usually written as y = lnx. (key just below log)read “y equals the natural log of x”Find: ln3 ln5 ln15The properties of common logs apply

to natural logs also. (Quotient, Power, Product)

http://www.algebra.com/algebra/homework/logarithm/Natural-Logarithms.lesson

Ex1) Use natural logs to solve 8e2x= 20.8e2x = 20e2x = 20/8e2x = 2.5

lne2x = ln2.52xlne = ln2.5

*ln(e) cancel each other out just like log(10)2x = ln2.5X = ln2.5/2

*Make sure you close parenthesis around top!

X = 0.458

Natural logs were invented because they simplify the expression of many relationships in the physical world…

p.338 Ex #2A rocket can attain a stable orbit 200 km above Earth if it reaches a velocity of 7.7 km/sec. The formula for the maximum velocity V of a rocket is given by V = -0.0098t + clnR. The rocket engines fire for t seconds and the velocity of the exhaust is c km/s. The ratio of the mass of the rocket with fuel to its mass without fuel is R. Suppose a rocket used to propel a spacecraft has a mass ratio of about 25, an exhaust velocity of 2.8 km/sec, and a firing time of 100 sec. Can this rocket attain a stable orbit?

V = -0.0098t + clnRV = -0.0098(100) + 2.8(ln)25

V = about 8.03 km/s

More than 7.7 km/s is needed for a stable orbit, so it can be attained.

#7) Find the velocity of a rocket with a mass ratio of about 15, an exhaust velocity of 2.1 km/sec, and a firing time of 30 sec. Can the rocket achieve a stable orbit 300 km above the earth?V = -0.0098t + clnRV = -0.0098(30) + 2.1ln(15)V = 5.39 km/secNo

Assignment

Section 7.6p.339-340

#14-1718-22 E24-26

(10 problems total)

Handouts:Half Life

Life Expectancy

Homework

Wrap-up: Due Thursdayp.343-345#1 - 50 all

Test Review ThursdayReminder: Test Friday

Warm-up 10/30/08The function y = 200(1.04)x models the first grade population y of an elementary school x years after the year 2000.

a) Graph the function on your graphing calculator. Adjust the viewing window:xmin=0,xmax=20;ymin=0;ymax=500;yscl=100

b) Estimate when the 1st grade pop. will = 250.

c) When will the pop. reach 325?d) Check your solution algebraically.

Interesting Facts• On a dewy morning, if you look at your shadow

in the grass, the dew drops shine light back to your eye creating a halo called a heilgenschein (German for halo.)

• The correct response to the Irish greeting, "Top of the morning to you," is "and the rest of the day to yourself."

• Giraffes have no vocal cords. • Joe DiMaggio had more home runs than

strikeouts during his career. • All porcupines float in water.

• A-1 Steak Sauce contains both orange peel and raisins. • Many northern parishes (counties) of Louisiana did not

agree with the Confederate movement. To show their disapproval, they changed their names. That's why there is a Union Parish, Jefferson Parish, etc.

• The Pentagon, in Arlington, Virginia, has twice as many bathrooms as is necessary. When it was built in the 1940s, the state of Virginia still had segregation laws requiring separate toilet facilities for blacks and whites.

• Residents of the island of Lesbos are Lesbosians, rather than Lesbians. (Of course, lesbians are called lesbians because Sappho was from Lesbos.)

• The Chinese ideogram for 'trouble' symbolizes 'two women living under one roof'.

• Go over wrap-up• Half life handout

• Financial Futures handout• Study for test

Warm-up 10/31/08

1) You put $1500 into an account earning 7% interest compounded continuously. How long will it be until you have $2000 in your bank account?

2) Evaluate: log4256

3) Expand: log4r2t4) Solve: 3lnx – ln2 = 45) Solve: 2logx = -46) Rewrite as a common log: log316

Interesting Facts

• German has a wood for the peace offerings brought to your mate when you've committed some conceived slight. This is "drachenfutter" or dragon's food.

• In Chinese, the words for crisis and opportunity are the same.

• No word in the English language rhymes with month.

• The poisonous copperhead smells like fresh cut cucumbers.

• In Disney's "Fantasia", the Sorcerer's name is "Yensid" (Disney backwards.)

• Anne Boleyn had six fingernails on one hand. • Mustard gas was invented in the McKinley Building on

the American University campus. Additionally, preliminary work on the Manhattan Project was done in that building. The government used the McKinley Building because of its unusual architecture. If there would be any type of large explosion inside the building, the building would implode onto itself, containing any lethal gas or nuclear material. The building now houses the Physics Department.

• When angered, the ears of Tasmanian devils turn a pinkish-red.

Questions?

After you finish the test: Work on make-up work,

Work on extra credit paper, Sleep?:P