Chapter 8 Exponential and Logarithmic Functions

8.1 Exponential Models

Exponential Functions An 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 B A is the y-intercept B is direction Growth Decay b > 1 0 < b < 1

Y-Intercept and Growth vs. Decay Identify each y-intercept and whether it is a growth or decay. Y= 3(1/4)x Y= .5(3)x 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

Write an exponential model for a graph that includes the points (2, 122.5) and (3, 857.5) (0, 24) and (3, 8/9)

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?

2(3)

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.

8.2 Exponential Functions

Exponential Function Where a = starting amount (y – intercept) b = change factor x = time

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?

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?

A population of 2500 triples in size every 10 years. What will the population be in 30 years?

50(1/2)^(1/14)x

Growth Decay . b > 1 0 < b < 1 (1 + r) (1 - r)

Percent to Change Factor Increase of 25% 2. Increase of 130% Decrease of 30% 4. Decrease of 80%

Growth Factor to Percent Find the percent increase or decease from the following exponential equations. y = 3(.5)x y = 2(2.3)x 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 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?

Compound Interest P = starting amount R = rate n = period T = time

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

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

8.3 Logarithmic Functions

Logarithmic Expressions Solve for x: 2x = 4 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 Logarithms To evaluate a log we are trying to “find the exponent.” Ex: Log5 25 Ask yourself: 5x = 25

You Try!

A Common Logarithm is a logarithm that uses base 10 A Common Logarithm is a logarithm that uses base 10. log 10 y = x ---- > log y = x Example: log1000

Common Log Find the Log: Log100 Log(1/10) The 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 equations y = 2x and y = log 2 x 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 Log 2 8 – log 2 4 = log 2 2 Log b x3y = 3(log b x) + log b y

Simplify Each Logarithm Log 3 20 – log 3 4 3(Log 2 x) + log 2 y 3(log 2) + log 4 – log 16

Expand Each Logarithm Log 5 (x/y) Log 3r4 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 Equations An Exponential Equation is an equation with an unknown for an exponent. Ex: 4x = 34

Try Some! 5x = 27 4. 3x+4 = 101 2. 73x = 20 5. 11x-5 + 50 = 250 3. 62x = 21

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

You Try! Log (7 – 2x) = -1 Log ( 5 – 2x) = 0 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! log 6 – log 3x = -2 log 5 – log 2x = 1

8.6 Natural Logarithms

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

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

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 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 form

Log to Exponential Form ln 1 = 0 ln 9 = 2.197 ln (5.28) = 1.6639

Simplify 3 ln 5 ln 5 + ln 4 ln 20 – ln 10 4 ln x + ln y – 2 ln z

Expand Ln (xy2) Ln(x/4) Ln(y/2x)

Solving Exponential Equations

Solving Logarithmic Equations Ln x = -2 Ln (2m + 3) = 8 1.1 + Ln x2 = 6

Homework PG 464 # 2 – 8, 14 – 28 (all even)