Post on 11-Jan-2016
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
-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
4.0
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
-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
4.0
-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
4.0
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
-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
4.0
-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
4.0
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
-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
4.0
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
-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
4.0
-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
4.0
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
-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
4.0
Exponential and Log Functions B. Davis MathScience Innovation Center
Compare Deals
-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
4.0
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 !
-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
4.0
Al
Braa
Gee
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.
-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
4.0
Y = 2^x
-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
4.0
Y = 3^x
-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
4.0
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.
-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
4.0
Y = 2^x
Y = .2^x
Y = .5^x
-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
4.0
-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
4.0
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
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
4.0
-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
4.0
-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
4.0
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
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
4.0
-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
4.0
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
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
4.0
-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
4.0
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
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
4.0
-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 5.0
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
4.0
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