1. Exponential and Logarithmic Functions MathScience Innovation Center Betsey Davis

2. Exponential and Log Functions B. Davis MathScience Innovation Center Great Offer ! w Your Uncle Al, Cousin Gee, and Auntie Braa each make you an offer you can’t refuse. w Each wants to give you $$$ every month until you graduate. w Your parents will only let you select one of the offers. w Which offer should you choose if each relative is increasing the size of the payments monthly?

3. Exponential and Log Functions B. Davis MathScience Innovation Center Here are the choices: w Uncle Al pays $1 the first month (June this year) and adds 2 additional dollars with every new monthly payment. w Cousin Gee pays 1 cent the first month (June this year) and doubles the payment every month. w 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.

4. Exponential and Log Functions B. Davis MathScience Innovation Center Al’s deal

5. Exponential and Log Functions B. Davis MathScience Innovation Center Al’s deal

6. Exponential and Log Functions B. Davis MathScience Innovation Center Gee’s Deal

7. Exponential and Log Functions B. Davis MathScience Innovation Center Gee’s Deal

8. Exponential and Log Functions B. Davis MathScience Innovation Center Braa’s Deal

9. Exponential and Log Functions B. Davis MathScience Innovation Center Braa’s Deal

10. Exponential and Log Functions B. Davis MathScience Innovation Center Compare Deals Al Braa Gee 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?

11. Exponential and Log Functions B. Davis MathScience Innovation Center Compare Deals AlBraaGee Are the results the same if we look at totals? Braa’s deal looks better after 5 months !

12. Exponential and Log Functions B. Davis MathScience Innovation Center Compare Deals Al Enter into TI 83 + List1: sequence to create 1,2,3,4,… 24 List 2: sequence to create 1,3,5,7,9...

13. Exponential and Log Functions B. Davis MathScience Innovation Center Compare Deals Gee Enter into TI 83 + List 3: sequence to create.01,.02,.04,.08, and so on...

14. Exponential and Log Functions B. Davis MathScience Innovation Center Compare Deals Braa Enter into TI 83 + List 4: sequence to create.50,2,4.5,8,12.5...

15. Exponential and Log Functions B. Davis MathScience Innovation Center Compare Deals AlBraaGee 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…. 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?

16. Exponential and Log Functions B. Davis MathScience Innovation Center Compare Equations AlBraaGee Al y = 2x -1 Gee y =.5x^2 Braa y =.005 *2^x Note different scale factors

17. Exponential and Log Functions B. Davis MathScience Innovation Center Let’s name the functions ! Al Braa Gee linear exponential quadratic

18. 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?

19. Exponential and Log Functions B. Davis MathScience Innovation Center Let’s look for patterns: w Uncle Al pays $1 the first month (June this year) and adds 2 additional dollars with every new monthly payment. w Cousin Gee pays 1 cent the first month (June this year) and doubles the payment every month. w 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. w Uncle Al pays $1 the first month (June this year) and adds 2 additional dollars with every new monthly payment. w Cousin Gee pays 1 cent the first month (June this year) and doubles the payment every month. w 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.

20. Exponential and Log Functions B. Davis MathScience Innovation Center Let’s look for patterns: w Uncle Al pays $1 the first month (June this year) and adds 2 additional dollars with every new monthly payment. w Cousin Gee pays 1 cent the first month (June this year) and doubles the payment every month. w 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. w Uncle Al pays $1 the first month (June this year) and adds 2 additional dollars with every new monthly payment. w Cousin Gee pays 1 cent the first month (June this year) and doubles the payment every month. w 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...

21. Exponential and Log Functions B. Davis MathScience Innovation Center Let’s look for patterns: w Uncle Al pays $1 the first month (June this year) and adds 2 additional dollars with every new monthly payment. w Cousin Gee pays 1 cent the first month (June this year) and doubles the payment every month. w 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. w Uncle Al pays $1 the first month (June this year) and adds 2 additional dollars with every new monthly payment. w Cousin Gee pays 1 cent the first month (June this year) and doubles the payment every month. w 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. Braa is adding…but increases the increasing amount steadily

22. Exponential and Log Functions B. Davis MathScience Innovation Center Let’s look for patterns: w Uncle Al pays $1 the first month (June this year) and adds 2 additional dollars with every new monthly payment. w Cousin Gee pays 1 cent the first month (June this year) and doubles the payment every month. w 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. w Uncle Al pays $1 the first month (June this year) and adds 2 additional dollars with every new monthly payment. w Cousin Gee pays 1 cent the first month (June this year) and doubles the payment every month. w 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.

23. Exponential and Log Functions B. Davis MathScience Innovation Center Let’s look for patterns: w Uncle Al pays $1 the first month (June this year) and adds 2 additional dollars with every new monthly payment. w Cousin Gee pays 1 cent the first month (June this year) and doubles the payment every month. w 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. w Uncle Al pays $1 the first month (June this year) and adds 2 additional dollars with every new monthly payment. w Cousin Gee pays 1 cent the first month (June this year) and doubles the payment every month. w 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.

24. Exponential and Log Functions B. Davis MathScience Innovation Center Let’s look for patterns: w Uncle Al pays $1 the first month (June 2003) and adds 2 additional dollars with every new monthly payment. w Cousin Gee pays 1 cent the first month (June 2003) and doubles the payment every month. w 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. w Uncle Al pays $1 the first month (June this year) and adds 2 additional dollars with every new monthly payment. w Cousin Gee pays 1 cent the first month (June this year) and doubles the payment every month. w 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..01,.02,.04,.08… is a geometric sequence.

25. Exponential and Log Functions B. Davis MathScience Innovation Center Exponential Functions w Variable is the exponent w base >0 w and base = 1. w y = b^x is the parent function. Y = 2^x Y = 3^x Y = 4^x

26. Exponential and Log Functions B. Davis MathScience Innovation Center What if 0<b<1 ? w Variable is the exponent w base >0 w and base = 1. w y = b^x is the parent function. Y = 2^x Y =.2^x Y =.5^x

27. Exponential and Log Functions B. Davis MathScience Innovation Center Summary of base y = b ^x w B is never negative w B is not 1 w when B is between 0 and 1, the function decreases always (decay ) w when B is bigger than 1, the function increases always (growth)

28. Exponential and Log Functions B. Davis MathScience Innovation Center Exponential Decay w Certain radioactive elements decay over time…. Half life is the time to decrease 1/2 of the amount. B 0. w This fraction is the rate of decrease.

29. Exponential and Log Functions B. Davis MathScience Innovation Center Exponential Growth w In nature, uninhibited, uncontrolled grow is exponential. B > 1 w This B is the rate of increase.

30. Exponential and Log Functions B. Davis MathScience Innovation Center Exponential Growth and Decay w More examples: w serum blood drug levels w atmospheric pressure w light absorption in seawater w compound interest growth w inflation rates

31. Exponential and Log Functions B. Davis MathScience Innovation Center Transformations of y = 2^x w Y = 2^x + 1 w moves up 1 w y = 2^x -1 w moves down 1

32. Exponential and Log Functions B. Davis MathScience Innovation Center Transformations of y = 2^x w Y = 2^(x + 1) w moves 1 left w y = 2^(x -1) w moves 1 right

33. Exponential and Log Functions B. Davis MathScience Innovation Center Transformations of y = 2^x w Y =3* 2^x w vertical stretch w y =.2*2^x w vertical shrink

34. Exponential and Log Functions B. Davis MathScience Innovation Center Transformations of y = 2^x w Y =-( 2^x) w flips over x w y = 2^(-x) w flips over y

35. Exponential and Log Functions B. Davis MathScience Innovation Center Solving exponential equations w Y = b ^x : 3 different unknowns Y = 2 ^3 y = 8 25 = 5 ^x x = 2 100 = b ^2 b= 10 This is the tricky one ! Just cube Just find square root

36. Exponential and Log Functions B. Davis MathScience Innovation Center Solving exponential equations 25 = 5 ^x x = 2 We need an inverse operation like squares and square roots 102 = 2 ^x ?

37. Exponential and Log Functions B. Davis MathScience Innovation Center Solving exponential equations 102 = 2 ^x ? Logarithms ( logs for short !) are the inverses of exponentials Log 2 102 = x

38. Exponential and Log Functions B. Davis MathScience Innovation Center Limitations of your calculator w It only knows log with base 10 and log with base e. w log = log with base 10 w ln = log with base e w To do other logs, use the change of base formula: y = log a b = log a / log b