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Exponential Functions Logarithmic Functions Compound Interest

5. Exponential Functions Logarithmic Functions Compound Interest Differentiation of Exponential Functions Exponential Functions as Mathematical Models. Exponential and Logarithmic Functions. 5.1. Exponential Functions. Exponential Function. The function defined by

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Exponential Functions Logarithmic Functions Compound Interest

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1. 5 • Exponential Functions • Logarithmic Functions • Compound Interest • Differentiation of Exponential Functions • Exponential Functions as Mathematical Models Exponential and Logarithmic Functions

2. 5.1 Exponential Functions

3. Exponential Function • The function defined by is called an exponential function with baseb and exponentx. • The domain of f is the set of all real numbers.

4. Example • The exponential function with base2 is the function with domain(–, ). • Find the values of f(x) for selected values of x follow:

5. Example • The exponential function with base2 is the function with domain(–, ). • Find the values of f(x) for selected values of x follow:

6. Laws of Exponents • Let a and b be positive numbers and let x and y be real numbers. Then,

7. Examples • Let f(x) = 22x – 1. Find the value of x for which f(x) = 16. Solution • We want to solve the equation 22x – 1= 16 = 24 • But this equation holds if and only if 2x – 1 = 4 giving x = . Example 2, page 331

8. Examples • Sketch the graph of the exponential function f(x) = 2x. Solution • First, recall that the domain of this function is the set of real numbers. • Next, putting x = 0 gives y = 20 = 1, which is the y-intercept. (There is no x-intercept, since there is no value of x for which y = 0) Example 3, page 331

9. Examples • Sketch the graph of the exponential function f(x) = 2x. Solution • Now, consider a few values for x: • Note that 2xapproaches zero as xdecreases without bound: • There is a horizontal asymptote at y = 0. • Furthermore, 2xincreases without bound when xincreases without bound. • Thus, the range of f is the interval(0, ). Example 3, page 331

10. Examples • Sketch the graph of the exponential function f(x) = 2x. Solution • Finally, sketch the graph: y 4 2 f(x) = 2x x –2 2 Example 3, page 331

11. Examples • Sketch the graph of the exponential function f(x) = (1/2)x. Solution • First, recall again that the domain of this function is the set of real numbers. • Next, putting x = 0 gives y = (1/2)0 = 1, which is the y-intercept. (There is no x-intercept, since there is no value of x for which y = 0) Example 4, page 332

12. Examples • Sketch the graph of the exponential function f(x) = (1/2)x. Solution • Now, consider a few values for x: • Note that (1/2)xincreases without bound when xdecreases without bound. • Furthermore, (1/2)xapproaches zero as xincreases without bound: there is a horizontal asymptote at y = 0. • As before, the range of f is the interval(0, ). Example 4, page 332

13. Examples • Sketch the graph of the exponential function f(x) = (1/2)x. Solution • Finally, sketch the graph: y 4 2 f(x) = (1/2)x x –2 2 Example 4, page 332

14. Examples • Sketch the graph of the exponential function f(x) = (1/2)x. Solution • Note the symmetry between the two functions: y 4 2 f(x) = 2x f(x) = (1/2)x x –2 2 Example 4, page 332

15. Properties of Exponential Functions • The exponential functiony = bx (b > 0, b≠ 1) has the following properties: • Its domain is (–, ). • Its range is (0, ). • Its graph passes through the point (0, 1) • It is continuous on (–, ). • It is increasing on (–, ) if b > 1 and decreasing on (–, ) if b < 1.

16. The Base e • Exponential functions to the basee, where e is an irrational number whose value is 2.7182818…, play an important role in both theoretical and applied problems. • It can be shown that

17. Examples • Sketch the graph of the exponential function f(x) = ex. Solution • Since ex> 0 it follows that the graph ofy = exis similar to the graph ofy = 2x. • Consider a few values for x: Example 5, page 333

18. Examples • Sketch the graph of the exponential function f(x) = ex. Solution • Sketching the graph: y f(x) = ex 5 3 1 x –3 –1 1 3 Example 5, page 333

19. Examples • Sketch the graph of the exponential function f(x) = e–x. Solution • Since e–x> 0 it follows that 0 < 1/e < 1 and so f(x) = e–x= 1/ex= (1/e)x is an exponential function with base less than1. • Therefore, it has a graph similar to that of y = (1/2)x. • Consider a few values for x: Example 6, page 333

20. Examples • Sketch the graph of the exponential function f(x) = e–x. Solution • Sketching the graph: y 5 3 1 f(x) = e–x x –3 –1 1 3 Example 6, page 333

21. 5.2 Logarithmic Functions

22. Logarithms • We’ve discussed exponential equations of the form y = bx (b > 0, b≠ 1) • But what about solving the same equation fory? • You may recall that y is called the logarithm of x to the base b, and is denoted logbx. • Logarithm of x to the base b y = logbxif and only ifx = by(x > 0)

23. Examples • Solve log3x= 4 for x: Solution • By definition, log3x= 4 implies x = 34 = 81. Example 2, page 338

24. Examples • Solve log164= x for x: Solution • log164= x is equivalent to 4 = 16x = (42)x= 42x, or 41 = 42x, from which we deduce that Example 2, page 338

25. Examples • Solve logx8= 3 for x: Solution • By definition, we see that logx8= 3 is equivalent to Example 2, page 338

26. Logarithmic Notation log x = log10xCommon logarithm ln x = logexNatural logarithm

27. Laws of Logarithms • If m and n are positive numbers, then

28. Examples • Given that log 2 ≈ 0.3010, log 3 ≈ 0.4771, and log 5 ≈ 0.6990, use the laws of logarithms to find Example 4, page 339

29. Examples • Given that log 2 ≈ 0.3010, log 3 ≈ 0.4771, and log 5 ≈ 0.6990, use the laws of logarithms to find Example 4, page 339

30. Examples • Given that log 2 ≈ 0.3010, log 3 ≈ 0.4771, and log 5 ≈ 0.6990, use the laws of logarithms to find Example 4, page 339

31. Examples • Given that log 2 ≈ 0.3010, log 3 ≈ 0.4771, and log 5 ≈ 0.6990, use the laws of logarithms to find Example 4, page 339

32. Examples • Expand and simplify the expression: Example 5, page 340

33. Examples • Expand and simplify the expression: Example 5, page 340

34. Examples • Expand and simplify the expression: Example 5, page 340

35. Logarithmic Function • The function defined by is called the logarithmic function with baseb. • The domain of f is the set of all positive numbers.

36. Properties of Logarithmic Functions • The logarithmic function y = logbx (b > 0, b≠ 1) has the following properties: • Its domain is (0, ). • Its range is (–, ). • Its graph passes through the point (1, 0). • It is continuous on (0, ). • It is increasing on (0, ) if b > 1 and decreasing on (0, ) if b < 1.

37. Example • Sketch the graph of the function y = ln x. Solution • We first sketch the graph of y = ex. • The required graph is the mirror image of the graph of y = ex with respect to the line y = x: y y = x y = ex y = ln x 1 x 1 Example 6, page 341

38. Properties Relating Exponential and Logarithmic Functions • Properties relating ex and ln x: eln x = x (x > 0) ln ex = x(for any real number x)

39. Examples • Solve the equation 2ex + 2 = 5. Solution • Divide both sides of the equation by 2 to obtain: • Take the natural logarithm of each side of the equation and solve: Example 7, page 342

40. Examples • Solve the equation 5 ln x + 3 = 0. Solution • Add –3 to both sides of the equation and then divide both sides of the equation by 5 to obtain: and so: Example 8, page 343

41. 5.3 Compound Interest

42. Compound Interest • Compound interest is a natural application of the exponential function to business. • Recall that simple interest is interest that is computed only on the original principal. • Thus, if I denotes the interest on a principalP (in dollars) at an interest rate of r per year for t years, then we have I = Prt • The accumulated amount A, the sum of the principal and interest after t years, is given by

43. Compound Interest • Frequently, interest earned is periodically added to the principal and thereafter earns interest itself at the same rate. This is called compound interest. • Suppose \$1000 (the principal) is deposited in a bank for a term of 3 years, earning interest at the rate of 8% per year compounded annually. • Using the simple interest formula we see that the accumulated amount after the first year is or \$1080.

44. Compound Interest • To find the accumulated amount A2at the end of the second year, we use the simple interest formulaagain, this time with P =A1, obtaining: or approximately \$1166.40.

45. Compound Interest • We can use the simple interest formulayet again to find the accumulated amount A3 at the end of the third year: or approximately \$1259.71.

46. Compound Interest • Note that the accumulated amounts at the end of each year have the following form: • These observations suggest the following general rule: • If P dollars are invested over a term of tyears earning interest at the rate of r per year compounded annually, then the accumulated amount is or:

47. Compounding More Than Once a Year • The formula was derived under the assumption that interest was compoundedannually. • In practice, however, interest is usually compoundedmore than once a year. • The interval of time between successive interest calculations is called the conversion period.

48. Compounding More Than Once a Year • If interest at a nominal a rate of r per year is compoundedm times a year on a principal of P dollars, then the simple interest rate per conversion period is • For example, the nominal interest rate is 8% per year, and interest is compounded quarterly, then or 2% per period.

49. Compounding More Than Once a Year • To find a general formula for the accumulated amount, we apply repeatedly with the interest rate i = r/m. • We see that the accumulated amount at the end of each period is as follows:

50. Compound Interest Formula • There are n = mt periods in t years, so the accumulated amount at the end oftyear is given by where A= Accumulated amount at the end of t years P= Principal r= Nominal interest rate per year m= Number of conversion periods per year t= Term (number of years)

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