LOGARITHMS AND SAVINGS ACCOUNTSWeek 8 LSP 120 Joanna Deszcz

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

For Example

log101000 = 3 because

log1010000 = 4 because

log1010,000,000 = 7

log1030 = 1.477 because

log10300 = 2.477 because

103 = 1000 104 = 10000 107 = 10,000,000 101.477 = 30 102.477 = 300

Log 10 X means… The Power to which 10 is raised to get X

So in general … log b X = Y because bY = X

Conversions

Should know how to convert from 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

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

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.

Petri dish continued…

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

Set up the problem then… Follow these steps

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

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

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 = 20.731 years

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!!

Savings Accounts

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…

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

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

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 = (1.00525)^4x

Take log of both side Log 2 = log(1.00525)^4x

Follow rule #2 Log 2= 4x* log(1.00525)

Divide by log(1.00525) 33.35 years to double your money

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

More on Calculating Interest

Check out this link ABC's of Figuring Interest