Lesson 10-5 Applications of Exponential and Logarithmic FunctionsObjective: To use exponential and logarithmic functions to solve problems.

Applications of exponential & logarithmic functions

Compound InterestContinuous CompoundingExponential Growth or decay (bacteria/ radiation half life)Richter Scale

Compound interestCompound interest means the each payment is calculated by including the interest previously earned on the investment.

Investing at 10% interest Compounded Annually

Year Investment at Start Interest Investment at End

0 (Now) $1,000.00 ($1,000.00 × 10% = ) $100.00 $1,100.00

1 $1,100.00 ($1,100.00 × 10% = ) $110.00 $1,210.00

2 $1,210.00 ($1,210.00 × 10% = ) $121.00 $1,331.00

3 $1,331.00 ($1,331.00 × 10% = ) $133.10 $1,464.10

4 $1,464.10 ($1,464.10 × 10% = ) $146.41 $1,610.51

5 $1,610.51

formula

If you have a bank account whose principal = $1000, and your bank compounds the interest twice a year at an interest rate of 5%, how much money do you have in your account at the year's end?

Solution

Continous Compounding

When n gets very large it approaches becoming continuous compounding. The formula is:

P = principal amount (initial investment)r = annual interest rate (as a decimal)t = number of yearsA = amount after time t 

rtPeA

Example

An amount of $2,340.00 is deposited in a bank paying an annual interest rate of 3.1%, compounded continuously. Find the balance after 3 years.Solution

A = 2340 e(.031)(3)

A = 2568.06

Exponential Growth

A = Pert   ...or... A = Pekt ...or... Q =ekt ...or... Q = Q0ekt

k is the growth constant

Bacteria Growth

In t hours the number of bacteria in a culture will grow to be approximately Q = Q0e2t where Q0 is the original number of bacteria. At 1 PM the culture has 50 bacteria. How many bacteria does it have at 4 PM? at noon?

Q = 50e2(3) Q = 50e2(-1)

Q = 50e6 Q = 50e-2

Q = 20,248 Q = 7

Practice

1. If you start a bank account with $10,000 and your bank compounds the interest quarterly at an interest rate of 8%, how much money do you have at the year’s end ? (assume that you do not add or withdraw any money from the account)2. An amount of $1,240.00 is deposited in a bank paying an annual interest rate of 2.85 %, compounded continuously. Find the balance after 2½ years.

Solution1.

2. A = 1240e(.0285)(2.5)

= $1,331.57

Warm up

The first credit card that you got charges 12.49 % interest to its customers and compounds that interest monthly. Within one day of getting your first credit card, you max out the credit limit by spending $1,200.00 . If you do not buy anything else on the card and you do not make any payments, how much money would you owe the company after 6 months? A = P(1 + )nt

n

r

Exponential Decay

An artifact originally had 12 grams of carbon-14 present. The decay model A = 12e-0.000121t

describes the amount of carbon-14 present after t years. How many grams of carbon-14 will be present in this artifact after 10,000 years?

A = 12e-0.000121t

A = 12e-0.000121(10,000)

A = 12e-1.21

A = 3.58

Earthquake – Richter scale

R = log It compares how much

stronger the earthquake is compared to a given standardR= 3.0 then 3 = log 1000 =

I = 1000I0 1000 times the standard

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Earthquake – Richter scale

Haiti 7.0 7 = log

10,000,000 =

Japan 8.9 8.9 = log 794,328,235 =

Virginia 5.9 ?(August 23, 2011)

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794,328