Transcript Logarithms and Savings Accounts

LOGARITHMS AND
SAVINGS ACCOUNTS
Week 8
LSP 120
Joanna Deszcz
What is a Logarithm?
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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
Log 10 X means…
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log101000 = 3 because
log1010000 = 4 because
log1010,000,000 = 7
log1030 = 1.477 because
log10300 = 2.477 because
So in general … log b X = Y
The Power to which 10 is raised to
get X
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103 = 1000
104 = 10000
107 = 10,000,000
101.477 = 30
102.477 = 300
because bY = X
Conversions
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Should know how to convert from
Log form to exponent (logb X = Y)
 Exponent form to log ( by = X)
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Notes:
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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
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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
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2 Important Log Rules
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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…
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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.
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Petri dish continued…
Set up the problem then…
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Use percent
increase/decrease
formula
= P * (1 + r)X
 Y = P * (1 - r)X
Y
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Equation for this situation
is 3000 = 100 * (1+.05)X
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We need to solve for X.
Follow these steps
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Step 1 - divide both sides by 100
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Step 2 - take the log of both
sides.
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Log 30 = log 1.05X
Step 3 - follow rule #2
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Result is 30 = 1.05X
Bring the X down in front
Log 30 = X * log 1.05.
Step 4 - divide both sides by log
1.05
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log (30) / log (1.05) = X
Use Excel to Calculate logs
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Built-in Log function in Excel
=log(number, base) or
 If
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no base is specified, 10 is assumed
=log10(number)
In this case…
 log
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(30) / log (1.05) = X
In Excel
 =log(30)/log(1.05)
 =69.71
minutes exactly to reach 3000 bacteria
Another Problem
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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
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The equation would be 200=100*(1+.034)X
Now solve for X
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Step 1 divide both sides by 100
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Step 2 - take the log of both sides
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Log 2 = log1.034X
Step 3 - follow rule #2
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2 = 1.034X
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
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Bacteria Again
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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
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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)
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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
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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
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What is your balance after 5 years?
How long would it take your money to double?
 Hint:
Use logs
Using Logs
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Use the percentage increase/decrease formula
In this case Y=P*(1+r)^x
 The equation?
 1600 = 800*(1+.021/4)^4x
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2 = (1.00525)^4x
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Follow rule #2
Log 2= 4x* log(1.00525)
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Take log of both side
Log 2 = log(1.00525)^4x
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Divide by 800
Divide by log(1.00525)
33.35 years to double your money
Annual Percentage Yield (APY)
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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.
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Formula
 Use
Percentage change formula for 2 consecutive years
 =(new-old)/old
 Change value to a %, show 2 decimal places
More on Calculating Interest
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Check out this link
ABC's of Figuring Interest