Chapter 8 Exponential and Logarithmic Functions

Exponential Functions

Exponential Functions An exponential function is a function with the general form y = abx a ≠ 0 and b > 0, and b ≠ 1 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 a model Write an exponential model for a graph that includes the points (2,122.5) and (3,857.5) (0,24) and (3, 8/9)

Writing Exponential Functions Cooling times for a cup of coffee at various temps.

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.

Exponential Functions

Exponential Function Remember: 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?

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

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

Increase and Decrease by percent Exponential Models can also be used to show an increase or decrease by a percentage.

Increase and Decrease by percent The rate of increase or decrease is a percent, we use a change factor/base of 1 + r or 1 – r. Growth Decay . b > 1 0 < b < 1 Change factor Change Factor (1 + r) (1 - r)

Percent to Change Factor When something grows or decays by a percent, we have to add or subtract it from one to find b. 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 Compound Interest is another type exponential function: Here P = starting amount R = rate n = period T = time

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

Compound Interest 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 = 16 2x = 10

Logarithmic Expression A Logarithm solves for the missing exponent! Exponential Form Logarithmic Form y = bx Logby = x

Exp to Log form Given the following Exponential Functions, Convert to Logarithmic Functions. 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

Try Some!

Common Log A Common Logarithm is a logarithm that uses base 10. Log 10 y = x -------- > Log y = x EX. Log1000

Common Log The Calculator will do a Common Log for us!! Find the Log Log1000 Log100 Log(1/10)

Change of Base Formula When the base of the log is not 10, we can use a Change of Base Formula to find Logs with our calculator!

Try Some! Find the following Logarithms using change of base formula

Log Graphs Graph the pair of equations y = 2x and y = log 2 x What do you notice??

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 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

Try Some! 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

Try Some! Log 6 – log 3x = -2 Log 2x + Log x = 11 Log 5 – Log 2x = 1

Homework Change!! PG 456 # 34 – 46 even

Classwork PG 112 # 11 – 30 # 41 – 49 Pick 12 (4 from each section)

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 e 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.

Natural Logarithmic We call a log with a base of 10 “Common Log” We can call a log with a base of e “Natural Log” Natural Log is denoted with the “LN” 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 Change PG 464 # 2 – 8, 14 – 28 (all even)