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Chapter 5: Exponential and Logarithmic Functions

Chapter 5: Exponential and Logarithmic Functions. 5.1 Inverse Functions 5.2 Exponential Functions 5.3 Logarithms and Their Properties 5.4 Logarithmic Functions 5.5 Exponential and Logarithmic Equations and Inequalities

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Chapter 5: Exponential and Logarithmic Functions

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  1. Chapter 5: Exponential and Logarithmic Functions 5.1 Inverse Functions 5.2 Exponential Functions 5.3 Logarithms and Their Properties 5.4 Logarithmic Functions 5.5 Exponential and Logarithmic Equations and Inequalities 5.6 Further Applications and Modeling with Exponential and Logarithmic Functions

  2. 5.6 Further Applications and Modeling with Exponential and Logarithmic Functions • Physical Science Applications: • A0 is some initial quantity • t represents time • k > 0 represents the growth constant, and k < 0 represents the decay constant

  3. 5.6 Exponential Decay Function Involving Radioactive Isotopes Example Nuclear energy derived from radioactive isotopes can be used to supply power to space vehicles. Suppose that the output of the radioactive power supply for a certain satellite is given by the function defined by where y is measured in watts and t is the time in days. • What is the initial output of the power supply? • After how many days will the output be reduced to 35 watts? (c) After how many days will the output be half of its initial amount? (That is, what is its half-life?)

  4. 5.6 Exponential Decay Function Involving Radioactive Isotopes Solution • Let t = 0 and evaluate y. The initial output is 40 watts. (b) Let y = 35 and solve for t.

  5. 5.6 Exponential Decay Function Involving Radioactive Isotopes (c) Because the initial amount is 40, the half-life is the value of t for which

  6. 5.6 Exponential Decay Function Involving Radioactive Isotopes The half-life can be obtained from the graph of by noting that when t = x = 173, y 20 =

  7. 5.6 Age of a Fossil using Carbon-14 Dating Example Carbon-14 is a radioactive form of carbon found in all living plants and animals. After a plant or animal dies, the radiocarbon disintegrates. Scientists determine the age of the remains by comparing the amount of carbon-14 present with the amount found in living plants and animals. The amount of carbon-14 present after t years is given by Find the half-life. Solution Let Divide by A0.

  8. 5.6 Age of a Fossil using Carbon-14 Dating Take the ln of both sides. The half-life is 5700 years. ln ex = x and quotient rule for logarithms Isolate t. Distribute and use the fact that ln1 = 0.

  9. 5.6 Finding Half-life Example Radium-226, which decays according to has a half-life of about 1612 years. Find k. How long does it take a 10-gram sample to decay to 6 grams? Solution The half-life tells us that A(1612) = (½)A0.

  10. 5.6 Finding Half-life Thus, radium-226 decays according to the equation Now let A(t) = 6 and A0 = 10 to find t.

  11. 5.6 Financial Applications Example How long will it take $1000 invested at 6%, compounded quarterly, to grow to $2700? Solution Find t when A = 2700, P = 1000, r = .06, and n = 4.

  12. 5.6 Amortization Payments • A loan of P dollars at interest i per period may be amortized in n equal periodic payments of R dollars made at the end of each period, where • The total interest I that will be paid during the term of the loan is

  13. 5.6 Using Amortization to Finance an Automobile Example You agree to pay $24,000 for a used SUV. After a down payment of $4000, the balance will be paid off in 36 equal monthly payments at 8.5% interest per year. Find the amount of each payment. How much interest will you pay over the life of the loan? Solution

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