Exponential and Logarithmic Functions

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

Gee’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

Exponential and Logarithmic Functions