Transcript Warm-up 5.1

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
Exponential
Growth
6
5
4
3
2
1
-6
-5
-4
-3
-2
-1
1
-1
-2
-3
-4
-5
2
3
4
5
6
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
Exponential
Decay
6
5
4
3
2
1
-6
-5
-4
-3
-2
-1
1
-1
-2
-3
-4
-5
2
3
4
5
6
Identify each as exponential growth or decay.
What is the percent increase/decrease?
1) Y = 1298(1.63)x
1) growth, 63%
2) Y = 2(0.65)x
2) Decay; 35%
3) Y = 12(1.7)x
3) Growth, 70%
4) Y = 16(0.25)
x
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 1994
Y = 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 7.1
p. 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 7.1
p. 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 Formula
A = P(1 + r/n)nt
P = principal
t = time in years
R = rate of interest
N = # 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)
Semi-Annually(2)
Quarterly (4)
Monthly (12)
1000(1+.1/1)1(1)
1000(1+.1/2)2(1)
1000(1+.1/4)4(1)
1100
Daily (365)
Hourly (8760)
By the Minute
(525,600)
1000(1+.1/12)12(1)
1000(1+.1/365)356(1)
1000(1+.1/8760)8760(1)
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=a
The 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-Life
Half 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 Formula
y = 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 technetium99m 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 hours
t = 75 hours
Amount 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 Example
1) 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 7.2
p.316
#16 – 22 Even
31 - 33,35 - 39
Warm-up
7.3
Identify 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 7.2
p. 316
#16 – 22 Even
31 - 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
3/20/09
1) 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
Exponentials
And
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 relationship
Exponential 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 5
45 = 1024
Exponent in a different form
1) log443= 3
43 = 43
2) log552 = x
5x = 52
3) log225 = y
2y = 25
y=5
4) logbb2 = z
bz = b2
z=2
From Logs to Exponents
1) 52 = 25
base is 5, exp is 2: log5(25) = 2
2) (1/2)3 = 1/8
log(1/2)(1/8) = 3
3) 24 = 64
log2(64) = 4
To evaluate logarithms, you can write
them in exponential form…
Evaluating Exponents
Ex. Log816
8x = 16
Both 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 x
X = 4/3
So, log816 = 4/3
Ex)
Evaluate log5125.
5x = 125
5x = 53
Since both bases are five, you can set
the exponents equal.
X=3
Thus, 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
b) Log41
c) Log31
d) Log51
2) What is the value of logbb?
a) log22
b) Log55
c) log44
d) log99
3) Explain why the base b in y = logbx
cannot equal 1.
Inverse Graphs
Logarithms 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 7.3
p.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 lefthanded.
• 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 7.3
p.321-323
#2 -12, 16 - 26,
30 - 40, 50 Even
Go Over
Questions?
Graphing Inverses
Complete math toolbox
p.324
#1 - 9 odd
Exponentials in the Real World
http://www.thefutureschannel.com/do
ckets/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 + logbN
Product Property
2) logbM/N = logbM – logbN
Quotient Property
3) logbMk= klogbM
Power 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 16
log(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/hom
ework/logarithm/Properties-ofLogarithms.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
Assignment
Section 7.4
P. 328 – 329
#2 – 34, 38 – 68
Even
Extra Practice?
http://wwwmath.cudenver.edu/~rbyrne/flash.htm
http://www.purplemath.com/modules/index
.htm
Warm-up
Rewrite each
expression
1) logx + logy
2) 2logx – logx
3) log3 + log4
Evaluate the
expression
4) log28 – log22
7.1-7.4
1) log(xy)
2) logx
3) 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 7.4
P. 328 – 329
#2 – 68
Multiples 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 = 31
x2/3 = (31 – 4)
x2/3= 27
Multiply both sides by reciprocal of
3/2
x(2/3)(3/2) = 273/2
x = 273/2
x=
Try these
1) 3y4/3 = 768
2) 6√(10y) = 3
3) √(7x) – 4 = 0
4) 102x =
1,000,000
5) Why is it
important to
check the
possible
solutions?
1) 64
2) 72.9
3) 16/7
4) 3
5) Squaring both
sides of an
equation can
lead to
extraneous
solutions.
Word Problem
Ex2. 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
hours
D3 = 216(144)2
D3 = 4478976
D = 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
Solve
2log9(√x) – log9(6x – 1) = 0
2log9(√x) = log9(6x – 1)
log9(√x)2= log9(6x – 1)
x= 6x – 1
1 = 5x
1/5 = x
Always plug the answer back in to
make sure that it won’t produce any
negatives or zeros in the logarithms.
Solve
logx + log(x – 1) = log(3x + 12)
log(x(x – 1)) = log(3x + 12)
x2 – x = 3x + 12
x2 – 4x - 12 = 0
(x – 6)(x + 2) = 0
X = 6, -2
When you plug them back in, 6 works,
but -2 is an extraneous solution.
Solve
ln10 – ln(7 – x) = lnx
ln10/(7 – x) = lnx
10 = x
7–x
10 = x(7 – x)
10 = 7x – x2
x2 – 7x + 10 = 0
(x – 5)(x – 2) = 0
x = 5, 2
Both are solutions.
Ex. 73x = 20
log73x = log20
Use rules of exponents to get:
3xlog7 = log20
Divide both sides by 3log7
x = log20/3log7
x = 0.513
Try these
1) 3x = 4
1) 1.262
2) 62x = 21
2) 0.850
3) 3x+4 = 101
3) 0.201
Sometimes it’s more
useful to convert an
equation to
exponential form to
work a problem.
More Practice
Solve
logx = 1 – log(x – 3)
logx + log(x – 3) = 1
log(x(x – 3) = 1
log(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 – 8x
x2 – 6x +8x = 8
x2 +2x - 8 = 0
(x + 4)(x – 2) = 0
x = -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 = 1500
log662x = log61500
2xlog66 = log61500
log66 = 1…so
2x = log1500/log6
base)
2x = 4.0816
x = 2.0408
(change of
Ex5
a) Solve log(3x + 1) = 5
log (101) undo each other = 1, so
10log(3x + 1) = 105
3x + 1 = 105
3x + 1 = 100,000
3x = 99,999
x = 33,333
Ex6
b) Solve 2logx – log3 = 2
use rules of exponents
log x2 = 2
3
101log(x2/3) = 102
x2/3 = 102
x2 = 102(3)
x2 = 100(3)
x = √300 = ±17.32
7.5 Assignment/HW
Section 7.5
p. 334 – 336
#5 – 60
(multiples of 5)
Warm-up
10/28/08
7.5 Lesson Quiz
• On ½ sheet of paper, complete &
solve the following 4 problems.
1) x3/2 = 27
2) 2 + 3x = 82
3) log x + log 4 = 3
4) 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 Paper
Due: 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?
Last
Section…
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
ln15
The properties of common logs apply
to natural logs also. (Quotient,
Power, Product)
http://www.algebra.com/algebra/hom
ework/logarithm/Natural-
Ex1) Use natural logs to solve 8e2x= 20.
8e2x = 20
e2x = 20/8
e2x = 2.5
lne2x = ln2.5
2xlne = ln2.5
*ln(e) cancel each other out just like log(10)
2x = ln2.5
X = 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 #2
A 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 + clnR
V = -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 + clnR
V = -0.0098(30) + 2.1ln(15)
V = 5.39 km/sec
No
Assignment
Section 7.6
p.339-340
#14-17
18-22 E
24-26
(10 problems total)
Handouts:
Half Life
Life Expectancy
Homework
Wrap-up: Due Thursday
p.343-345
#1 - 50 all
Test Review Thursday
Reminder: Test Friday
Warm-up
10/30/08
The 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: log4r2t
4) Solve: 3lnx – ln2 = 4
5) Solve: 2logx = -4
6) 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