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## Exponential and Logarithmic Functions

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**Exponential and LogarithmicFunctions**MathScience Innovation Center Betsey Davis**Great Offer !**• Your Uncle Al, Cousin Gee, and Auntie Braa each make you an offer you can’t refuse. • Each wants to give you $$$ every month until you graduate. • Your parents will only let you select one of the offers. • Which offer should you choose if each relative is increasing the size of the payments monthly? Exponential and Log Functions B. Davis MathScience Innovation Center**Here are the choices:**• Uncle Al pays $1 the first month (June this year) and adds 2 additional dollars with every new monthly payment. • Cousin Gee pays 1 cent the first month (June this year) and doubles the payment every month. • Auntie Braa pays 50 cents the first month (June this year), and adds$1.50 more to the2nd payment, $2.50 more to the 3rd month, $3.50 more to the 4th month, and so on. Exponential and Log Functions B. Davis MathScience Innovation Center**Al’s deal**Exponential and Log Functions B. Davis MathScience Innovation Center**Al’s deal**Exponential and Log Functions B. Davis MathScience Innovation Center**Gee’s Deal**Exponential and Log Functions B. Davis MathScience Innovation Center**Gee’s Deal**Exponential and Log Functions B. Davis MathScience Innovation Center**Braa’s Deal**Exponential and Log Functions B. Davis MathScience Innovation Center**Braa’s Deal**Exponential and Log Functions B. Davis MathScience Innovation Center**Compare Deals**Al Gee Braa Which is better at the end of 1 month? Which is better at the end of 2 months? Which is better at the end of 3 months? Are the results the same if we look at totals? Exponential and Log Functions B. Davis MathScience Innovation Center**Al**Gee Braa Compare Deals Braa’s deal looks better after 5 months ! Are the results the same if we look at totals? Exponential and Log Functions B. Davis MathScience Innovation Center**Al**Compare Deals Enter into TI 83 + List1: sequence to create 1,2,3,4,… 24 List 2: sequence to create 1,3,5,7,9... Exponential and Log Functions B. Davis MathScience Innovation Center**Gee**Compare Deals Enter into TI 83 + List 3: sequence to create .01,.02,.04,.08, and so on... Exponential and Log Functions B. Davis MathScience Innovation Center**Braa**Compare Deals Enter into TI 83 + List 4: sequence to create .50,2,4.5,8,12.5... Exponential and Log Functions B. Davis MathScience Innovation Center**Al**Gee Braa Compare Deals Who gives biggest monthly payment in the very beginning? Do one of the other two catch up to him/her and when? Does the third person ever catch up and when? Turn on STAT PLOTS: Plot 1 list 1 and list 2 Plot 2 list 1 and list 3 Plot 3 list 1 and list 4 Adjust window…. Exponential and Log Functions B. Davis MathScience Innovation Center**Al**Gee Braa Compare Equations Note different scale factors Al y = 2x -1 Gee y = .5x^2 Braa y = .005 *2^x Exponential and Log Functions B. Davis MathScience Innovation Center**Let’s name the functions !**Al linear Gee exponential Braa quadratic Exponential and Log Functions B. Davis MathScience Innovation Center**Let’s look at total money…**Create “cumsum” lists for Al, Gee, and Braa When does Gee’s total payment become the best deal? Exponential and Log Functions B. Davis MathScience Innovation Center**Let’s look for patterns:**• Uncle Al pays $1 the first month (June this year) and adds 2 additional dollars with every new monthly payment. • Cousin Gee pays 1 cent the first month (June this year) and doubles the payment every month. • Auntie Braa pays 50 cents the first month (June this year), and adds$1.50 more to the2nd payment, $2.50 more to the 3rd month, $3.50 more to the 4th month, and so on. • Uncle Al pays $1 the first month (June this year) and adds 2 additional dollars with every new monthly payment. • Cousin Gee pays 1 cent the first month (June this year) and doubles the payment every month. • Auntie Braa pays 50 cents the first month (June this year), and adds$1.50 more to the2nd payment, $2.50 more to the 3rd month, $3.50 more to the 4th month, and so on. Exponential and Log Functions B. Davis MathScience Innovation Center**Let’s look for patterns:**• Uncle Al pays $1 the first month (June this year) and adds 2 additional dollars with every new monthly payment. • Cousin Gee pays 1 cent the first month (June this year) and doubles the payment every month. • Auntie Braa pays 50 cents the first month (June this year), and adds$1.50 more to the2nd payment, $2.50 more to the 3rd month, $3.50 more to the 4th month, and so on. • Uncle Al pays $1 the first month (June this year) and adds 2 additional dollars with every new monthly payment. • Cousin Gee pays 1 cent the first month (June this year) and doubles the payment every month. • Auntie Braa pays 50 cents the first month (June this year), and adds$1.50 more to the2nd payment, $2.50 more to the 3rd month, $3.50 more to the 4th month, and so on. Al is steadily increasing by adding a constant amount…linear…. Arithmetic sequence1,3,5,7... Exponential and Log Functions B. Davis MathScience Innovation Center**Braa is adding…but increases the increasing amount**steadily Let’s look for patterns: • Uncle Al pays $1 the first month (June this year) and adds 2 additional dollars with every new monthly payment. • Cousin Gee pays 1 cent the first month (June this year) and doubles the payment every month. • Auntie Braa pays 50 cents the first month (June this year), and adds$1.50 more to the2nd payment, $2.50 more to the 3rd month, $3.50 more to the 4th month, and so on. • Uncle Al pays $1 the first month (June this year) and adds 2 additional dollars with every new monthly payment. • Cousin Gee pays 1 cent the first month (June this year) and doubles the payment every month. • Auntie Braa pays 50 cents the first month (June this year), and adds$1.50 more to the2nd payment, $2.50 more to the 3rd month, $3.50 more to the 4th month, and so on. Exponential and Log Functions B. Davis MathScience Innovation Center**Let’s look for patterns:**• Uncle Al pays $1 the first month (June this year) and adds 2 additional dollars with every new monthly payment. • Cousin Gee pays 1 cent the first month (June this year) and doubles the payment every month. • Auntie Braa pays 50 cents the first month (June this year), and adds$1.50 more to the2nd payment, $2.50 more to the 3rd month, $3.50 more to the 4th month, and so on. Sequence..but not arithmetic .5, 2, 4.5 ,8 , 12.5,… these are each 1/2 of perfect squares. • Uncle Al pays $1 the first month (June this year) and adds 2 additional dollars with every new monthly payment. • Cousin Gee pays 1 cent the first month (June this year) and doubles the payment every month. • Auntie Braa pays 50 cents the first month (June this year), and adds$1.50 more to the2nd payment, $2.50 more to the 3rd month, $3.50 more to the 4th month, and so on. Exponential and Log Functions B. Davis MathScience Innovation Center**Let’s look for patterns:**• Uncle Al pays $1 the first month (June this year) and adds 2 additional dollars with every new monthly payment. • Cousin Gee pays 1 cent the first month (June this year) and doubles the payment every month. • Auntie Braa pays 50 cents the first month (June this year), and adds$1.50 more to the2nd payment, $2.50 more to the 3rd month, $3.50 more to the 4th month, and so on. • Uncle Al pays $1 the first month (June this year) and adds 2 additional dollars with every new monthly payment. • Cousin Gee pays 1 cent the first month (June this year) and doubles the payment every month. • Auntie Braa pays 50 cents the first month (June this year), and adds$1.50 more to the2nd payment, $2.50 more to the 3rd month, $3.50 more to the 4th month, and so on. Gee is multiplying his payment by a steady amount, 2. Exponential and Log Functions B. Davis MathScience Innovation Center**Let’s look for patterns:**• Uncle Al pays $1 the first month (June this year) and adds 2 additional dollars with every new monthly payment. • Cousin Gee pays 1 cent the first month (June this year) and doubles the payment every month. • Auntie Braa pays 50 cents the first month (June this year), and adds$1.50 more to the2nd payment, $2.50 more to the 3rd month, $3.50 more to the 4th month, and so on. • Uncle Al pays $1 the first month (June 2003) and adds 2 additional dollars with every new monthly payment. • Cousin Gee pays 1 cent the first month (June 2003) and doubles the payment every month. • Auntie Braa pays 50 cents the first month (June 2003), and adds$1.50 more to the2nd payment, $2.50 more to the 3rd month, $3.50 more to the 4th month, and so on. .01, .02, .04, .08… is a geometric sequence. Exponential and Log Functions B. Davis MathScience Innovation Center**Y = 2^x**Exponential Functions • Variable is the exponent • base >0 • and base = 1. • y = b^x is the parent function. Y = 3^x Y = 4^x Exponential and Log Functions B. Davis MathScience Innovation Center**Y = 2^x**What if 0<b<1 ? • Variable is the exponent • base >0 • and base = 1. • y = b^x is the parent function. Y = .2^x Y = .5^x Exponential and Log Functions B. Davis MathScience Innovation Center**Summary of base y = b ^x**• B is never negative • B is not 1 • when B is between 0 and 1, the function decreases always (decay ) • when B is bigger than 1, the function increases always (growth) Exponential and Log Functions B. Davis MathScience Innovation Center**Exponential Decay**• Certain radioactive elements decay over time…. Half life is the time to decrease 1/2 of the amount. B< 1 but B>0. • This fraction is the rate of decrease. Exponential and Log Functions B. Davis MathScience Innovation Center**Exponential Growth**• In nature, uninhibited, uncontrolled grow is exponential. B > 1 • This B is the rate of increase. Exponential and Log Functions B. Davis MathScience Innovation Center**Exponential Growth and Decay**• More examples: • serum blood drug levels • atmospheric pressure • light absorption in seawater • compound interest growth • inflation rates Exponential and Log Functions B. Davis MathScience Innovation Center**Transformations of y = 2^x**• Y = 2^x + 1 • moves up 1 • y = 2^x -1 • moves down 1 Exponential and Log Functions B. Davis MathScience Innovation Center**Transformations of y = 2^x**• Y = 2^(x + 1) • moves 1 left • y = 2^(x -1) • moves 1 right Exponential and Log Functions B. Davis MathScience Innovation Center**Transformations of y = 2^x**• Y =3* 2^x • vertical stretch • y = .2*2^x • vertical shrink Exponential and Log Functions B. Davis MathScience Innovation Center**Transformations of y = 2^x**• Y =-( 2^x) • flips over x • y = 2^(-x) • flips over y Exponential and Log Functions B. Davis MathScience Innovation Center**Solving exponential equations**• Y = b ^x : 3 different unknowns • Y = 2 ^3 • y = 8 • 25 = 5 ^x • x = 2 Just cube Just find square root • 100 = b ^2 • b= 10 This is the tricky one ! Exponential and Log Functions B. Davis MathScience Innovation Center**Solving exponential equations**We need an inverse operation like squares and square roots • 25 = 5 ^x • x = 2 102 = 2 ^x ? Exponential and Log Functions B. Davis MathScience Innovation Center**Solving exponential equations**Logarithms ( logs for short !) are the inverses of exponentials 102 = 2 ^x ? Log2 102 = x Exponential and Log Functions B. Davis MathScience Innovation Center**Limitations of your calculator**• It only knows log with base 10 and log with base e. • log = log with base 10 • ln = log with base e • To do other logs, use the change of base formula: y = logab = log a / log b Exponential and Log Functions B. Davis MathScience Innovation Center