Chapter 8 Exponential and Logarithmic Functions

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Chapter 8 Exponential and Logarithmic Functions

+8.1 Exponential Models

+Exponential FunctionsAn exponential function is a function with the general form

y = abx

Graphing Exponential Functions What does a do? What does b do?1. y = 3( ½ )x  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

+A and BA is the y-interceptB is directionGrowth Decayb > 1 0 < b < 1

+Y-Intercept and Growth vs. DecayIdentify 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 EDITSTAT CALC 0:ExpReg

+ Write an exponential model for a graph that includes the points

1. (2, 122.5) and (3, 857.5)

2. (0, 24) and (3, 8/9)

+Modeling Exponential FunctionsSuppose 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?

+IntervalsWhen 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.

+8.2 Exponential Functions

+Exponential Function

Wherea = starting amount (y – intercept)b = change factorx = time

y=abx

+Modeling Exponential FunctionsSuppose 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?

+Modeling Exponential FunctionsSuppose 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?

+A population of 2500 triples in size every 10 years.

What will the population be in 30 years?

+Growth Decay .

b > 1 0 < b < 1

(1 + r) (1 - r)

+Percent to Change Factor1. Increase of 25% 2. Increase of

130%

2. Decrease of 30% 4. Decrease of 80%

+Growth Factor to PercentFind 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?

+Percent Increase and DecreaseThe 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?

+Compound Interest

P = starting amountR = raten = periodT = time

+Compound InterestFind the balance of a checking account that has $3,000 compounded annually at 14% for 4 years.P = R = n = T =

+Compound InterestFind the balance of a checking account that has $500 compounded semiannually at 8% for 5 years.P = R = n = T =

+8.3 Logarithmic Functions

+Logarithmic ExpressionsSolve for x:1. 2x = 4

2. 2x = 10

+Logarithmic Expression

A Logarithm solves for the missing exponent:

Exponential Form

Logarithmic Form

y = bx logby = x

+Convert the following exponential functions to logarithmic Functions.1. 42 = 16

2. 51 = 5

3. 70 = 1

+Log to Exp form

Given the following Logarithmic Functions, Convert to Exponential Functions.1. Log4 (1/16) = -2 2. Log255 = ½

+Evaluating LogarithmsTo evaluate a log we are trying to “find the exponent.”

Ex: Log5 25 Ask yourself: 5x = 25

+ You Try!

log2 32log3 81log36 6log71log2 8log16 4

1.

2.

3.

4.

5.

6.

+A Common Logarithm is a logarithm that uses base 10.

log 10 y = x ---- > log y = x

Example: log1000

+Common LogThe Calculator will do a Common Log for us!Find the Log:Log100

Log(1/10)

+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 equations1. y = 2x and y = log 2 x

2. y = 3x and y = log 3 x

What do you notice??

+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

+8.4 Properties of Logarithms

+Properties of Logs

Product Property loga(MN)=logaM + logaN

Quotient Property loga(M/N)=logaM – logaN

Power Property Loga(Mp)=p*logaM

+Identify the Property

1. Log 2 8 – log 2 4 = log 2 2

2. Log b x3y = 3(log b x) + log b y

+Simplify Each Logarithm1. Log 3 20 – log 3 4

2. 3(Log 2 x) + log 2 y

3. 3(log 2) + log 4 – log 16

+Expand Each Logarithm1. Log 5 (x/y)

2. Log 3r4

3. Log 2 7b

+8.5 Exponential and Logarithmic Equations

+Remember!

Exponential and Logarithmic equations are INVERSES of one another.

Because of this, we can use them to solve each type of equation!

+Exponential EquationsAn Exponential Equation is an equation with an unknown for an exponent.Ex: 4x = 34

+Try Some!1. 5x = 27

2. 73x = 20

3. 62x = 21

4. 3x+4 = 101

5. 11x-5 + 50 = 250

+Logarithmic Equation

To Solve Logarithmic Equation we can transform them into Exponential Equations!Ex: Log (3x + 1) = 5

+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

+You Try!1. log 6 – log 3x = -2

2. log 5 – log 2x = 1

+8.6 Natural Logarithms

+Compound InterestFind the balance in an account paying 3.2% annual interest on $10,000 in 18 years compounded quarterly.

+The Constant: ee 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

+

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.

+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

+Exponential to Log form1. ex = 6

2. ex = 25

3. ex + 5 = 32

+Log to Exponential Form 1. ln 1 = 0

2. ln 9 = 2.197

3. ln (5.28) = 1.6639

+Simplify1. 3 ln 5

2. ln 5 + ln 4

3. ln 20 – ln 10

4. 4 ln x + ln y – 2 ln z

+Expand1. Ln (xy2)

2. Ln(x/4)

3. Ln(y/2x)

+Solving Exponential Equations1. ex = 18

2. ex+1 = 30

3. e2x = 12

+Solving Logarithmic Equations 1. Ln x = -2

2. Ln (2m + 3) = 8

3. 1.1 + Ln x2 = 6

+Homework

PG 464 # 2 – 8, 14 – 28 (all even)