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Lesson 10-5 Applications of Exponential and Logarithmic Functions

Lesson 10-5 Applications of Exponential and Logarithmic Functions. Objective: To use exponential and logarithmic functions to solve problems. Applications of exponential & logarithmic functions. Compound Interest Continuous Compounding

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Lesson 10-5 Applications of Exponential and Logarithmic Functions

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

  2. Applications of exponential & logarithmic functions • Compound Interest • Continuous Compounding • Exponential Growth or decay (bacteria/ radiation half life) • Richter Scale

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

  4. Investing at 10% interest Compounded Annually

  5. 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?

  6. Solution

  7. 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

  8. 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

  9. Exponential Growth • A = Pert...or...   A = Pekt   ...or...   Q =ekt • ...or...   Q = Q0ekt • k is the growth constant

  10. 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

  11. 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.

  12. Solution • 1. • 2. A = 1240e(.0285)(2.5) = $1,331.57

  13. 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

  14. 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

  15. Earthquake – Richter scale • R = log It compares how much • stronger the earthquake is compared to a given standard • R= 3.0 then 3 = log 1000 = I = 1000I0 1000 times the standard

  16. 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) 794,328

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