1. © 2008 Pearson Addison-Wesley. All rights reserved 8-6-1 Chapter 1 Section 8-6 Exponential and Logarithmic Functions, Applications, and Models

2. © 2008 Pearson Addison-Wesley. All rights reserved 8-6-2 Exponential and Logarithmic Functions, Applications, and Models Exponential Functions and Applications Logarithmic Functions and Applications Exponential Models in Nature

3. © 2008 Pearson Addison-Wesley. All rights reserved 8-6-3 Exponential Function An exponential function with base b, where b > 0 and is a function of the form where x is any real number.

4. © 2008 Pearson Addison-Wesley. All rights reserved 8-6-4 Example: Exponential Function (b > 1) y x The x-axis is the horizontal asymptote of each graph. x2 x2 x –11/2 01 12 24

5. © 2008 Pearson Addison-Wesley. All rights reserved 8-6-5 Example: Exponential Function (0 < b < 1) y x The x-axis is the horizontal asymptote of each graph. x(1/2) x –24 –12 01 11/2

6. © 2008 Pearson Addison-Wesley. All rights reserved 8-6-6 Graph of 1. The graph always will contain the point (0, 1). 2. When b > 1 the graph will rise from left to right. When 0 < b < 1, the graph will fall from left to right. 3. The x-axis is the horizontal asymptote. 4. The domain is and the range is

7. © 2008 Pearson Addison-Wesley. All rights reserved 8-6-7 Exponential with Base e y x

8. © 2008 Pearson Addison-Wesley. All rights reserved 8-6-8 Compound Interest Formula Suppose that a principal of P dollars is invested at an annual interest rate r (in percent, expressed as a decimal), compounded n times per year. Then the amount A accumulated after t years is given by the formula

9. © 2008 Pearson Addison-Wesley. All rights reserved 8-6-9 Example: Compound Interest Formula Suppose that $2000 dollars is invested at an annual rate of 8%, compounded quarterly. Find the total amount in the account after 6 years if no withdrawals are made. Solution

10. © 2008 Pearson Addison-Wesley. All rights reserved 8-6-10 Example: Compound Interest Formula Solution (continued) There would be $3216.88 in the account at the end of six years.

11. © 2008 Pearson Addison-Wesley. All rights reserved 8-6-11 Continuous Compound Interest Formula Suppose that a principal of P dollars is invested at an annual interest rate r (in percent, expressed as a decimal), compounded continuously. Then the amount A accumulated after t years is given by the formula

12. © 2008 Pearson Addison-Wesley. All rights reserved 8-6-12 Example: Continuous Compound Interest Formula Suppose that $2000 dollars is invested at an annual rate of 8%, compounded continuously. Find the total amount in the account after 6 years if no withdrawals are made. Solution

13. © 2008 Pearson Addison-Wesley. All rights reserved 8-6-13 Example: Compound Interest Formula Solution (continued) There would be $3232.14 in the account at the end of six years.

14. © 2008 Pearson Addison-Wesley. All rights reserved 8-6-14 Exponential Growth Formula The continuous compound interest formula is an example of an exponential growth function. In situations involving growth or decay of a quantity, the amount present at time t can often be approximated by a function of the form where A 0 represents the amount present at time t = 0, and k is a constant. If k > 0, there is exponential growth; if k < 0, there is exponential decay.

15. © 2008 Pearson Addison-Wesley. All rights reserved 8-6-15 Definition of For b > 0,

16. © 2008 Pearson Addison-Wesley. All rights reserved 8-6-16 Exponential and Logarithmic Equations Exponential Equation Logarithmic Equation

17. © 2008 Pearson Addison-Wesley. All rights reserved 8-6-17 Logarithmic Function A logarithmic function with base b, where b > 0 and is a function of the form

18. © 2008 Pearson Addison-Wesley. All rights reserved 8-6-18 Graph of The graph of y = log b x can be found by interchanging the roles of x and y in the function f (x) = b x. Geometrically, this is accomplished by reflecting the graph of f (x) = b x about the line y = x. The y-axis is called the vertical asymptote of the graph.

19. © 2008 Pearson Addison-Wesley. All rights reserved 8-6-19 Example: Logarithmic Functions y x y x

20. © 2008 Pearson Addison-Wesley. All rights reserved 8-6-20 Graph of 1. The graph always will contain the point (1, 0). 2. When b > 1 the graph will rise from left to right. When 0 < b < 1, the graph will fall from left to right. 3. The y-axis is the vertical asymptote. 4. The domain is and the range is

21. © 2008 Pearson Addison-Wesley. All rights reserved 8-6-21 Natural Logarithmic Function g(x) = ln x, called the natural logarithmic function, is graphed below. y x

22. © 2008 Pearson Addison-Wesley. All rights reserved 8-6-22 Natural Logarithmic Function The expression ln e k is the exponent to which the base e must be raised in order to obtain e k. There is only one such number that will do this, and it is k. Thus for all real numbers k,

23. © 2008 Pearson Addison-Wesley. All rights reserved 8-6-23 Example: Doubling Time Suppose that a certain amount P is invested at an annual rate of 5% compounded continuously. How long will it take for the amount to double (doubling time)? Solution Sub in 2P for A (double). Divide by P.

24. © 2008 Pearson Addison-Wesley. All rights reserved 8-6-24 Example: Doubling Time Solution (continued) Therefore, it would take about 13.9 years for the initial investment P to double. Divide by.05 Take ln of both sides. Simplify.

25. © 2008 Pearson Addison-Wesley. All rights reserved 8-6-25 Exponential Models in Nature Radioactive materials disintegrate according to exponential decay functions. The half- life of a quantity that decays exponentially is the amount of time it takes for any initial amount to decay to half its initial value.

26. © 2008 Pearson Addison-Wesley. All rights reserved 8-6-26 Example: Half-Life Carbon 14 is a radioactive form of carbon that is found in all living plants and animals. After a plant or animal dies, the radiocarbon disintegrates. The amount of carbon 14 present after t years is modeled by the exponential equation a) What is the half-life of carbon 14? b) If an initial sample contains 1 gram of carbon 14, how much will be left in 10,000 years?

27. © 2008 Pearson Addison-Wesley. All rights reserved 8-6-27 Example: Half-Life Solution a) The half-life of carbon 14 is about 5700 years.

28. © 2008 Pearson Addison-Wesley. All rights reserved 8-6-28 Example: Half-Life Solution b) There will be about.30 grams remaining.