Download Chapter 8 Exponential and Logarithmic Functions

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Chapter 8 Exponential
and Logarithmic
Functions
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8.1 Exponential Models
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Exponential Functions
An exponential function is a function with the
general form
y = abx
Graphing Exponential Functions
What does a do?
1. y = 3( ½ )x
What does b do?
2. y = 3( 2)x
3. y = 5( 2)x
4. y = 7( 2)x
5. y = 2( 1.25 )x
6. y = 2( 0.80 )x
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A and B
A is the y-intercept
B is direction
Growth
Decay
b>1
0<b<1
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Y-Intercept and Growth vs. Decay
Identify each y-intercept and whether it is a
growth or decay.
1.
Y= 3(1/4)x
2.
Y= .5(3)x
3.
Y = (.85)x
+ Writing Exponential Functions
Write an exponential model for a
graph that includes the points (2,2)
and (3,4).
STAT  EDIT
STAT  CALC  0:ExpReg
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Write an exponential model for a graph that
includes the points
1.
(2, 122.5) and (3, 857.5)
1.
(0, 24) and (3, 8/9)
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Modeling Exponential Functions
Suppose 20 rabbits are taken to an island. The
rabbit population then triples every year.
The function f(x) = 20 • 3x where x is the number
of years, models this situation.
What does “a” represent in this problems? “b”?
How many rabbits would there be after 2 years?
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Intervals
When something grows or decays at a
particular interval, we must multiply x by
the intervals’ reciprocal.
EX: Suppose a population of 300 crickets
doubles every 6 months.
Find the number of crickets after 24
months.
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8.2 Exponential Functions
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Exponential Function
y  ab
x
Where
a = starting amount (y – intercept)
b = change factor
x = time
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Modeling Exponential Functions
Suppose a Zombie virus has infected 20
people at our school. The number of
zombies doubles every hour. Write an
equation that models this.
How many zombies are there after 5 hours?
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Modeling Exponential Functions
Suppose a Zombie virus has infected 20
people at our school. The number of
zombies doubles every 30 minutes. Write an
equation that models this.
How many zombies are there after 5 hours?
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A population of 2500 triples in size
every 10 years.
What
will the population be in 30
years?
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Growth
Decay
b>1
0<b<1
(1 + r)
(1 - r)
.
+ Percent to Change Factor
1.
Increase of 25%
2. Increase of 130%
1.
Decrease of 30%
4. Decrease of 80%
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Growth Factor to Percent
Find the percent increase or decease from
the following exponential equations.
1.
y = 3(.5)x
2.
y = 2(2.3)x
3.
y = 0.5(1.25)x
+ Percent Increase and Decrease
A dish has 212 bacteria in it. The population
of bacteria will grow by 80% every day.
How many bacteria will be present in 4 days?
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Percent Increase and Decrease
The house down the street has termites in the
porch. The exterminator estimated that
there are about 800,000 termites eating at
the porch. He said that the treatment he put
on the wood would kill 40% of the termites
every day.
 How
many termites will be eating at the porch
in 3 days?
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Compound Interest
æ rö
A = Pç1+ ÷
è nø
nt
P = starting amount
R = rate
n = period
T = time
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Compound Interest
Find the balance of a checking
account that has $3,000 compounded
annually at 14% for 4 years.
P=
R=
n=
T=
+ Compound Interest
Find the balance of a checking
account that has $500 compounded
semiannually at 8% for 5 years.
P=
R=
n=
T=
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8.3 Logarithmic Functions
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Logarithmic Expressions
Solve for x:
1.
2x = 4
1.
2x = 10
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Logarithmic Expression
A Logarithm solves for the missing exponent:
Exponential
Form
Logarithmic
Form
y = bx
logby = x
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Convert the following exponential functions to
logarithmic Functions.
1. 42 = 16
2. 51 = 5
3. 70 = 1
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Log to Exp form
Given the following Logarithmic Functions,
Convert to Exponential Functions.
1. Log4 (1/16) = -2
2. Log255 = ½
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Evaluating Logarithms
To evaluate a log we are trying to “find the
exponent.”
Ex: Log5 25
Ask yourself: 5x = 25
+ You Try!
1.
log 2 32
2.
log 3 81
3.
log 36 6
4.
log 7 1
5.
log 2 8
6.
log16 4
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A Common Logarithm is a logarithm that
uses base 10.
log 10 y = x
---- >
log y = x
Example:
log1000
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Common Log
The Calculator will do a Common Log for us!
Find the Log:
Log100
Log(1/10)
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When the base of the log is not 10, we can use
a Change of Base Formula to find Logs with our
calculator:
+ You Try!
Find the following Logarithms using change
of base formula
+ Graph the pair of equations
1.
y = 2x
and y = log 2 x
1.
y = 3x
and y = log 3 x
What do you notice??
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Graphing Logarithmic Functions
A logarithmic function is the inverse of an
exponential function.
The inverse of a function is the same as
reflecting a function across the line y = x
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8.4 Properties of Logarithms
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Properties of Logs
Product Property
loga(MN)=logaM + logaN
Quotient Property
loga(M/N)=logaM – logaN
Power Property
Loga(Mp)=p*logaM
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Identify the Property
1.
Log 2 8 – log 2 4 = log 2 2
2.
Log b x3y = 3(log b x) + log b y
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Simplify Each Logarithm
1.
Log 3 20 – log 3 4
1.
3(Log 2 x) + log 2 y
2.
3(log 2) + log 4 – log 16
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Expand Each Logarithm
1.
Log 5 (x/y)
2.
Log 3r4
3.
Log 2 7b
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8.5 Exponential and
Logarithmic Equations
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Remember!
Exponential and Logarithmic equations
are INVERSES of one another.
Because of this, we can use them to
solve each type of equation!
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Exponential Equations
An Exponential Equation is an equation
with an unknown for an exponent.
Ex:
4x = 34
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Try Some!
1.
5x = 27
2. 73x = 20
3. 62x = 21
4. 3x+4 = 101
5. 11x-5 + 50 = 250
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Logarithmic Equation
To Solve Logarithmic Equation we can
transform them into Exponential Equations!
Ex: Log (3x + 1) = 5
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You Try!
1.
Log (7 – 2x) = -1
2.
Log ( 5 – 2x) = 0
3.
Log (6x) – 3 = -4
+ Using Properties to Solve Equations
Use the properties of logs to simplify
logarithms first before solving!
Ex: 2 log(x) – log (3) = 2
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You Try!
1.
log 6 – log 3x = -2
1.
log 5 – log 2x = 1
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8.6 Natural Logarithms
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Compound Interest
Find the balance in an account paying
3.2% annual interest on $10,000 in 18
years compounded quarterly.
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The Constant: e
e is a constant very similar to π.
Π = 3.141592654…
e = 2.718281828…
Because it is a fixed number, we can find:
e2
e3
e4
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Exponential Functions with a base of e
are used to describe CONTINUOUS
growth or decay.
Some accounts compound interest, every
second. We refer to this as continuous
compounding.
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Continuously Compounded
Find the balance in an account paying 3.2%
annual interest on $10,000 in 18 years
compounded continuously.
Investment: You put $2000 into an account earning
4% interest compounded continuously. Find the
amount at the end of 8 years.
If $5,000 is invested in a savings account that pays 7.85%
interest compounded continuously, how much money
will be in the account after 12 years?
+ Natural Logarithms
-Log
with a base of 10: “Common Log”
-Log
with a base of e: “Natural Log” (ln)
- The natural logarithm of a number x is the power to
which e would have to be raised to equal x
Note: All the same rules and properties apply to
natural log as they do to regular logs
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Exponential to Log form
1.
ex = 6
2. ex = 25
3. ex + 5 = 32
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Log to Exponential Form
1.
ln 1 = 0
2.
ln 9 = 2.197
3.
ln (5.28) = 1.6639
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Simplify
1.
3 ln 5
2.
ln 5 + ln 4
3.
ln 20 – ln 10
1.
4 ln x + ln y – 2 ln z
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Expand
1.
Ln (xy2)
1.
Ln(x/4)
1.
Ln(y/2x)
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Solving Exponential Equations
1.
ex = 18
2.
ex+1 = 30
3.
e2x = 12
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Solving Logarithmic Equations
1.
Ln x = -2
2.
Ln (2m + 3) = 8
3.
1.1 + Ln x2 = 6
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Homework
PG 464 # 2 – 8, 14 – 28 (all even)