1. Exponential and Logarithm Functions
Growth: interest, births
Decay: isotopes, drug levels, temperature
Scales: Richter, pH, decibel levels
Functions involved are called exponential and logarithmic.

2. Exponential Functions
A function when the base(a) is some positive number.
The exponent is variable(x).
The exponential function with base a is defined by:

3. Example 1 Domain: Range: Horizontal Asymptote: x -2 -1 1 2 f(x) 1/41 2
f(x)
1/4
1/2 4
f (x) 4
Domain: 2
Range: x -1 1 2
Horizontal Asymptote:

4. Example 2 Domain: Range: Horizontal Asymptote: x -2 -1 1 2 f(x) 9 31 2
f(x) 9 3
1/3
1/9 9 3
Domain: x -1 1 2
Range:
Horizontal Asymptote:

5. Natural Base, e Special base, e  2.7182818……..
Use a calculator to evaluate the following values of the natural exponential function (round to 5 decimal places):

6. Logarithmic Functions
Exponential functions f (x) = ax are one-to-one functions.
This means they each have an inverse function, a function that reverses what the original function did.
We denote the inverse function with loga, the logarithmic function with base a, written as:
and we say “f of x is the logarithm of x base a”.

7. Exponential vs. logarithmic form
Switch from logarithmic form to exponential form:

8. Evaluating logarithms
Switch from exponential form to logarithmic form:

9. Graph Domain: Range: Vertical Asymptote:x 1
Vertical Asymptote:
Domain restrictions (from first week):
No negatives under an even root
No division by zero
Only positives inside a logarithm

10. Properties of logarithms and exponentials
1. logaax = x
(you must raise a to the power of x to get ax)
2. alogax = x
(logax is the power to which a must be raised to get x)
Both are also the result of composing a function with its inverse.

11. Common Logarithm (Base 10)
With calculator:
Without calculator:

12. Natural Logarithm (Base e)
With calculator:
Without calculator:
To evaluate other bases on the calculator, use the following change of base formula:
loga b

13. Solving equations
Isolate exponential function and apply logarithm function to both sides of the equation.
Isolate the logarithm function and apply the base to both sides of the equation.
Remember inverse properties and change of base:

14. Example 1

15. Example 2

16. Example 3

17. Example 4 (a) What is the initial number of bacteria?
(b) What is the relative rate of growth?
Express your answer as a percentage.
(c) How many bacteria are in the culture after 5 hours?
Please round the answer to the nearest integer.
(d) When will the number of bacteria reach 10,000?
Please round the answer to the nearest hundredth.

18. Example 5
The mass m(t) remaining after t days from 40 gram sample of polonium-210 is given by:
(a) How much remains after 60 days?
(b) When will 10 grams remain?
Please round the answer to the nearest day.
(c) Find the half-life of polonium-210.

19. Simple Interest and Sequences
I = interest
P = principal
r = rate
t = time
Future Value of investment: S = P + I
Ex. 1: $800 is invested for 5 years at an annual rate of 14%.
How much interest is earned?
What is future value?

20. Simple Interest (continued)
Ex. 2: If you borrow $1600 for 2 years at an annual rate of 14% simple interest, how much must you pay back?
How much interest is owed?
What do you pay back?
Ex. 3: $800 is invested at an annual rate of 14%.
How much is earned in 5 months?
How long does it take your investment to be worth $2200?

21. Sequences
A list of numbers following a certain pattern
a1 , a2 , a3 , a4 , … , an , …
Pattern is determined by position or by what has come before
3, 6, 12, , , …

22. Defined by n (its position)
Find the first four terms and the 100th term for the following:
Partial Sums
Adding the first n terms of a sequence, the nth partial sum:

23. Example 2 Find the first 4 partial sums and then the
nth partial sum for the sequence defined by:

24. Arithmetic Sequences 4, 7, 10, 13, 16, … 81, 75, 69, 63, 57, …
Consider the following sequences:
4, 7, 10, 13, 16, …
81, 75, 69, 63, 57, …
An arithmetic sequence is defined as:
with a as the first term and d as the common difference.

25. Examples

26. Partial Sums
Find the 1000th partial sum for arithmetic sequence with a=1, d=1:

27. Partial Sums
Find the 7th partial sum for arithmetic sequence with a=10, d=7:

28. Example
For the month of February, you get $4 on the first day, $7 on the second day, $10 on the third day and so on. How much will you have received in total at the end of the month?

29. Compound Interest and Sequences
S = future worth
P = principal
r = annual rate
t = time in years
m = number of compoundings per year
if compounded continuously.
When compounded more than once a year, the effective
annual rate (or annual percentage yield) is:

30. Example m S What would a $5000 investment be worth in 3 years
if the interest rate is 7.5% and the investment is compounded: m S
yearly
semiannually
monthly
continuously
Effective annual rate for
the monthly investment:

31. Fill in the table

32. Geometric Sequences Consider the following sequences:
3, 6, 12, 24, 48, …
81, 27, 9, 3, 1, …
A geometric sequence is defined as:
with a as the first term and r as the common ratio.

33. Examples
Find the common ratio, the nth term, and the 20th term.

34. Partial Sums
The partial sum of a geometric sequence looks like:

35. Example
Find the 20th partial sum for geometric sequence with a = 5, r = 2:

36. Preview of Annuities
Make yearly payments of $200 to an account that has an annual interest rate of 5%. What is the account worth after the 10th payment is made?

37. Future Worth of Annuities
Annuity – a sum of money paid in equal payments (the future worth of the annuity, S, is the sum of all payments plus any interest accrued.)

38. Example
Annuity Due – a sum of money paid in equal payments at the beginning of each time period.

39. Three Friends (aged 25) when they’re 60
Three friends from college take vastly different approaches
to saving for retirement:
Anne: Put away $5000 every year for 10 yrs, then sit on it for 25 yrs
Barry: Don’t save for 10 yrs, then save $5000 every year for the next 25 yrs
Clyde: Don’t save until turning 50, then play catch-up for 10 years until equal with Anne’s future worth at 60.
Their retirement account options all have a 6% annual rate
during the next 35 years.

40. Anne and Barry saved what by age 60
Anne: Put away $5000 every year for 10 yrs, then sit on it for 25 yrs
Barry: Don’t save for 10 yrs, then save $5000 every year for the next 25 yrs

41. And Clyde? Clyde: Don’t save until turning 50, then play
catch-up for 10 years until equal with Anne’s
future worth at 60.
How much must he deposit every year for 10 years?
Who’s made the best choice?

42. And then there’s Dara
Dara: Like Clyde, won’t save until turning 50, but then will put away
$10,000 every year until equal with Anne’s future worth at 60.
How long must she do this?

43. Present Worth of Annuities
make equal payments
Future worth of annuity (lump sum)
(plus interest)
Present worth of annuity (lump sum)
withdraw equal amounts
(while earning interest)
The present value of an annuity that pays out n regular payments of $R at the end of each time period from an account with an interest rate of i per time period is:

44. Example 1
From a settlement, you will be awarded $1000 at the end of every month for the next 30 years paid from an annuity earning interest at an annual rate of 6%.
What amount must be in the account to achieve this goal?

45. Annuity Due
Present Value of Annuity Due – when the payouts occur at the beginning of each time period.
Example 2
$1,260,000 Jackpot winner gets a choice of:
a) $3500/month for 30 years paid at the beginning of each month with 7.5% annual rate
b) Lump sum of the present value

46. Example 3
Tom and Alice inherit $100,000 and put this into an annuity paying out at the start of each month for 5 years with interest of 6.3%, compounded monthly. How much will they receive each month?
If they get an annuity paying 7.2%, how long will the same payout last?

47. Loans
Paying back a borrowed amount (An)in n regular equal payments(R), with interest rate i per time period is a form of present value annuity. Rewrite present value equation to get the size of each payment.

48. Example 1
You plan to borrow $20,000 for 4 years. What is the monthly payment if the current APR for car loans is 6%?
How much are you paying in interest for this loan?
in interest for this loan.

49. Example 2
A couple takes out a 30 year mortgage to borrow $ at 4.2%, compounded monthly. What is their monthly mortgage payment?
Amortization schedule details payment breakdown.

50. Example 2 - continued in interest. in interest.
If the couple pays an extra $100 per month,
how much would they save in interest?
Currently paying:
Now paying:
in interest.
in interest.