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Section 6.7 – Exponential & Logarithmic models

Section 6.7 – Exponential & Logarithmic models. Exponential Growth & Decay. Exponential and logarithmic functions with the base of e are often used in applications in business, science and sociology. The exponential function

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Section 6.7 – Exponential & Logarithmic models

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  1. Section 6.7 – Exponential & Logarithmic models

  2. Exponential Growth & Decay Exponential and logarithmic functions with the base of e are often used in applications in business, science and sociology. The exponential function , represents continuous exponential growth, while the exponential function , represents continuous exponential decay.

  3. Population Growth Growth Rate Initial Population Population 1. The population of a small town is 5,687 in 2005. The continuousgrowth rate is 6% per year. Find the exponential growth function.

  4. Population Growth • 2. Population Growth of Rabbits Under ideal conditions, a population of rabbits has an continuousgrowth rate of 11.7% per day. Consider an initial population of 100 rabbits. • a. Find the exponential growth function. • b. What will the population be after 7 days? • After 2 weeks? 2 weeks  t = 14

  5. Population Growth The doubling time is the amount of time it takes for the initial population to double. c. Find the doubling time. Get the exponential by itself. Divide both sides by 100. Write in logarithmic form. Solve for t. After approximately 5.9 days, you will have 200 rabbits.

  6. Interest Compounded Continuously Continuous Interest Rate Initial Investment Future Value 3. You invest $4,000 at an annual interest rate of 2.5% compounded continuously. Find the exponential growth function.

  7. Interest Compounded Continuously 4. Interest Compounded Continuously Suppose that $10,000 is invested at an interest rate of 5.4% per year, compounded continuously. a. Find the exponential function that describes the amount in the account after time t, in years. b. What is the balance after 1 yr? After 5 yrs?

  8. Interest Compounded Continuously c. What is the doubling time? Get the exponential by itself. Divide both sides by 10000. Write in logarithmic form. Solve for t. After approximately 12.8 years, you will have $20,000.

  9. Exponential Decay Decay Rate This model can be used in carbon dating and in the decay of radioactive substances. Initial Amount Amount of substance 5. We have a sample of Krypton-85 which has a decay rate of 6.3% per year. Find the exponential decay function.

  10. Carbon Decay 6. Carbon Dating The amount of carbon-14 present in animal bones after t years is given by , where is the amount of carbon present after t years. Find the age of a mummy discovered in the pyramid Khufu in Egypt, if it has lost 46% of the carbon-14. If it has lost 46% of its carbon, then it contain 54%. So the amount of carbon present in the mummy is 54% of its initial amount. Get the exponential by itself. Divide both sides by P0. Write in logarithmic form. The mummy is approximately 5,135 yrs. old. Solve for t.

  11. Half-Life The half-life of a substance is the amount of time it takes for of the initial amount of the substance to decay. That is, the amount of time it takes to become .

  12. Half-Life 7. Radioactive Decay The decay rate of Iodine-131 is 9.6% per day. What is its half-life? Write in logarithmic form or write each side as the argument of a logarithm. The half-life of Iodine-131 is approximately 7.2 days. Solve for t.

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