1. Logarithms and Savings Accounts
Week 8 LSP 120 Joanna Deszcz

2. What is a Logarithm?
A logarithm (or log) is a number that represents a power or exponent
Why use logs?
A simpler way to express large values
Some things grow or shrink exponentially, so the log is a perfect “numbering” system

3. For Example log101000 = 3 because log1010000 = 4 because
Log 10 X means…
The Power to which 10 is raised to get X
log = because
log = 4 because
log1010,000,000 = 7
log1030 = because
log10300 = because
103 = 1000
104 = 10000
107 = 10,000,000
= 30
= 300
So in general … log b X = Y because bY = X

4. Conversions Should know how to convert from Notes:
Log form to exponent (logb X = Y)
Exponent form to log ( by = X)
Notes:
When dealing with logs of base 10
when the X value is multiplied by 10
the outcome of the log or Y increases by 1
Logs written without the base number
Assumed to be base 10
We will be dealing only with log base 10
So log is the same as log10

5. 2 Important Log Rules 1. log 10x = x 2. log ax = x * log a
This rule used with exponential functions
Logs used to solve for exponent variable

6. Some Problems…
A Petri dish contains 100 bacteria cells. The number of cells increases 5% every minute.
We had previously asked the question, “How long will it take for the number of cells in the dish to reach 3000?”
Originally solved by setting up a table in Excel and found that after 69 minutes, there were 2898 cells and at 70 minutes, there were 3043 cells.
Obviously somewhere between minute 69 and 70, the number of cells reached 3000.
Using logarithms we can get an exact answer.

7. Petri dish continued… Use percent increase/decrease formula
Set up the problem then…
Follow these steps
Use percent increase/decrease formula
Y = P * (1 + r)X
Y = P * (1 - r)X
Equation for this situation is 3000 = 100 * (1+.05)X
We need to solve for X.
Step 1 - divide both sides by 100
Result is 30 = 1.05X
Step 2 - take the log of both sides.
Log 30 = log 1.05X
Step 3 - follow rule #2
Bring the X down in front
Log 30 = X * log 1.05.
Step 4 - divide both sides by log 1.05
log (30) / log (1.05) = X

8. Use Excel to Calculate logs
Built-in Log function in Excel
=log(number, base) or
If no base is specified, 10 is assumed
=log10(number)
In this case…
log (30) / log (1.05) = X
In Excel
=log(30)/log(1.05)
=69.71 minutes exactly to reach 3000 bacteria

9. Another Problem
If $100 in savings account grows at 3.4% compounded annually, how long will it take for the balance to double?
What would the equation be?
Solve

10. The Answer The equation would be 200=100*(1+.034)X Now solve for X
Step 1 divide both sides by 100
2 = 1.034X
Step 2 - take the log of both sides
Log 2 = log1.034X
Step 3 - follow rule #2
Log 2 = X* log 1.034
Step 4 - divide both sides by log 1.034
X = log (2) / log (1.034)
Use Excel to find that X = years

11. Bacteria Again
Assuming we allowed the bacteria population to reach 3000 then put in an antibiotic that killed the cells at a rate of 22.5% a minute, how long would it take for the population to decline to 60 cells?
What is the equation?
Solve on your own!!

12. Savings Accounts

13. Earning Interest in a Savings Account
Putting your money into a savings account is like loaning the bank your money
Buying savings bonds you actually loan money to the government
In return the bank/government pays you interest…
And gets to use your savings to generate more money
Through investments, loans, etc…

14. Annual Percentage Rate (APR)
The amount of interest you are paid for loaning your money
Formula for calculating APR is
A=P*(1+r/n)^(nY)
P = beginning balance
r = annual interest rate
n = compounding frequency (1=annually, 4 = quarterly, 12 = monthly)
Y = number of years

15. Example
You deposit $800 into a savings account that has and annual percentage rate of 2.1% compounded quarterly.
What is your balance after the first year?
A=P*(1+r/n)^nY)
A=800*(1+.021/4)^(4*1)
A=$816.93
What is your balance after 5 years?
How long would it take your money to double?
Hint: Use logs

16. Using Logs Use the percentage increase/decrease formula
In this case Y=P*(1+r)^x
The equation?
1600 = 800*(1+.021/4)^4x
Divide by 800
2 = ( )^4x
Take log of both side
Log 2 = log( )^4x
Follow rule #2
Log 2= 4x* log( )
Divide by log( )
33.35 years to double your money

17. Annual Percentage Yield (APY)
Percentage rate reflecting the total interest to be earned based on:
the interest rate
an institution’s compounding method
assuming funds remain in account for a 365-day year.
Formula
Use Percentage change formula for 2 consecutive years
=(new-old)/old
Change value to a %, show 2 decimal places

18. More on Calculating Interest
Check out this link
ABC's of Figuring Interest