Exponential and LogarithmicFunctions

MathScience Innovation Center

Betsey Davis

Exponential and Log Functions B. Davis MathScience Innovation Center

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

month payment Total money

1 1 1 2 3 4 3 5 9 4 7 16 5 9 25 6 11 36 7 13 49 8 15 64 9 17 81 10 19 100

Al’s deal

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Exponential and Log Functions B. Davis MathScience Innovation Center

month payment Total money

1 1 1 2 3 4 3 5 9 4 7 16 5 9 25 6 11 36 7 13 49 8 15 64 9 17 81 10 19 100

Al’s deal

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Exponential and Log Functions B. Davis MathScience Innovation Center

Gee’s Dealmonth payment Total

money 1 .01 .01 2 .02 .03 3 .04 .07 4 .08 .15 5 .16 .31 6 .32 .63 7 .64 1.27 8 1.28 2.55 9 2.56 5.11 10 5.12 10.23

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Exponential and Log Functions B. Davis MathScience Innovation Center

Gee’s Dealmonth payment Total

money 1 .01 .01 2 .02 .03 3 .04 .07 4 .08 .15 5 .16 .31 6 .32 .63 7 .64 1.27 8 1.28 2.55 9 2.56 5.11 10 5.12 10.23

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Exponential and Log Functions B. Davis MathScience Innovation Center

Braa’s Dealmonth payment Total

money 1 .50 .50 2 2.00 2.50 3 4.50 7.00 4 8.00 15.00 5 12.50 27.50 6 18.00 45.50 7 24.50 70.00 8 32.00 102.00 9 40.50 142.50 10 50.00 192.50

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Exponential and Log Functions B. Davis MathScience Innovation Center

Braa’s Dealmonth payment Total

money 1 .50 .50 2 2.00 2.50 3 4.50 7.00 4 8.00 15.00 5 12.50 27.50 6 18.00 45.50 7 24.50 70.00 8 32.00 102.00 9 40.50 142.50 10 50.00 192.50

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Exponential and Log Functions B. Davis MathScience Innovation Center

Compare Deals

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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?

Exponential and Log Functions B. Davis MathScience Innovation Center

Compare DealsAl BraaGee

month payment Total money

1 1 1 2 3 4 3 5 9 4 7 16 5 9 25 6 11 36 7 13 49 8 15 64 9 17 81 10 19 100

month payment Total money

1 .01 .01 2 .02 .03 3 .04 .07 4 .08 .15 5 .16 .31 6 .32 .63 7 .64 1.27 8 1.28 2.55 9 2.56 5.11 10 5.12 10.23

Are the results the same if we look at totals?

Braa’s deal looks better after 5 months !

Exponential and Log Functions B. Davis MathScience Innovation Center

Compare DealsAl

month payment Total money

1 1 1 2 3 4 3 5 9 4 7 16 5 9 25 6 11 36 7 13 49 8 15 64 9 17 81 10 19 100

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

Compare DealsGee

month payment Total money

1 .01 .01 2 .02 .03 3 .04 .07 4 .08 .15 5 .16 .31 6 .32 .63 7 .64 1.27 8 1.28 2.55 9 2.56 5.11 10 5.12 10.23

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

Compare DealsBraa

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

Compare DealsAl BraaGee

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?

Exponential and Log Functions B. Davis MathScience Innovation Center

Compare EquationsAl BraaGee

Al y = 2x -1

Gee y = .5x^2

Braa y = .005 *2^x

Note

different

scale factors

Exponential and Log Functions B. Davis MathScience Innovation Center

Let’s name the functions !

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linear

exponential

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

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.

Braa is adding…but increases the increasing amount steadily

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.

Sequence..but not arithmetic

.5, 2, 4.5 ,8 , 12.5,… these are each 1/2 of perfect squares.

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

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.

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.

.01, .02, .04, .08…

is a geometric sequence.

Exponential and Log Functions B. Davis MathScience Innovation Center

Exponential Functions

Variable is the exponent

base >0 and base = 1. y = b^x is the

parent function.

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Y = 2^x

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Y = 3^x

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Y = 4^x

Exponential and Log Functions B. Davis MathScience Innovation Center

What if 0<b<1 ?

Variable is the exponent

base >0 and base = 1. y = b^x is the

parent function.

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Y = 2^x

Y = .2^x

Y = .5^x

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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 -4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0

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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 -4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0

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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 -4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0

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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 -4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0

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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

•100 = b ^2

•b= 10This is the tricky one !

Just cubeJust find

square root

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 ?

Exponential and Log Functions B. Davis MathScience Innovation Center

Solving exponential equations

102 = 2 ^x ?

Logarithms ( logs for short !)

are the inverses of exponentials

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