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 GrowthDecay b > 10 < 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?
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 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?
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. GrowthDecay . b > 10 < 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.
Logarithmic Expressions Solve for x: • 2x = 4 • 2x = 16 • 2x = 10
Logarithmic Expression A Logarithm solves for the missing exponent!
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
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 • y = 3x and y = log 3 x What do you notice??
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
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 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
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