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# Lesson 13: Exponential and Logarithmic Functions (Section 041 slides)

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Definitions and elementary properties of exponential and logarithmic functions.

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### Lesson 13: Exponential and Logarithmic Functions (Section 041 slides)

1. 1. Sections 3.1–3.2 Exponential and Logarithmic Functions V63.0121.041, Calculus I New York University October 20, 2010 Announcements Midterm is graded and scores are on blackboard. Should get it back in recitation. There is WebAssign due Monday/Tuesday next week. . . . . . .
2. 2. Announcements Midterm is graded and scores are on blackboard. Should get it back in recitation. There is WebAssign due Monday/Tuesday next week. . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 2 / 37
3. 3. Objectives for Sections 3.1 and 3.2 Know the definition of an exponential function Know the properties of exponential functions Understand and apply the laws of logarithms, including the change of base formula. . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 3 / 37
4. 4. Outline Definition of exponential functions Properties of exponential Functions The number e and the natural exponential function Compound Interest The number e A limit Logarithmic Functions . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 4 / 37
5. 5. Derivation of exponential functions Definition If a is a real number and n is a positive whole number, then an = a · a · · · · · a n factors . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 5 / 37
6. 6. Derivation of exponential functions Definition If a is a real number and n is a positive whole number, then an = a · a · · · · · a n factors Examples 23 = 2 · 2 · 2 = 8 34 = 3 · 3 · 3 · 3 = 81 (−1)5 = (−1)(−1)(−1)(−1)(−1) = −1 . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 5 / 37
7. 7. Anatomy of a power Definition A power is an expression of the form ab . The number a is called the base. The number b is called the exponent. . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 6 / 37
8. 8. Fact If a is a real number, then ax+y = ax ay (sums to products) ax ax−y = y a (ax )y = axy (ab)x = ax bx whenever all exponents are positive whole numbers. . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 7 / 37
9. 9. Fact If a is a real number, then ax+y = ax ay (sums to products) ax ax−y = y (differences to quotients) a (ax )y = axy (ab)x = ax bx whenever all exponents are positive whole numbers. . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 7 / 37
10. 10. Fact If a is a real number, then ax+y = ax ay (sums to products) ax ax−y = y (differences to quotients) a (ax )y = axy (repeated exponentiation to multiplied powers) (ab)x = ax bx whenever all exponents are positive whole numbers. . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 7 / 37
11. 11. Fact If a is a real number, then ax+y = ax ay (sums to products) ax ax−y = y (differences to quotients) a (ax )y = axy (repeated exponentiation to multiplied powers) (ab)x = ax bx (power of product is product of powers) whenever all exponents are positive whole numbers. . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 7 / 37
12. 12. Fact If a is a real number, then ax+y = ax ay (sums to products) ax ax−y = y (differences to quotients) a (ax )y = axy (repeated exponentiation to multiplied powers) (ab)x = ax bx (power of product is product of powers) whenever all exponents are positive whole numbers. . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 7 / 37
13. 13. Fact If a is a real number, then ax+y = ax ay (sums to products) ax ax−y = y (differences to quotients) a (ax )y = axy (repeated exponentiation to multiplied powers) (ab)x = ax bx (power of product is product of powers) whenever all exponents are positive whole numbers. Proof. Check for yourself: ax+y = a · a · · · · · a = a · a · · · · · a · a · a · · · · · a = ax ay x + y factors x factors y factors . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 7 / 37
14. 14. Let's be conventional The desire that these properties remain true gives us conventions for ax when x is not a positive whole number. . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 8 / 37
15. 15. Let's be conventional The desire that these properties remain true gives us conventions for ax when x is not a positive whole number. For example, what should a0 be? We cannot write down zero a’s and multiply them together. But we would want this to be true: ! an = an+0 = an · a0 . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 8 / 37
16. 16. Let's be conventional The desire that these properties remain true gives us conventions for ax when x is not a positive whole number. For example, what should a0 be? We cannot write down zero a’s and multiply them together. But we would want this to be true: ! ! an an = an+0 = an · a0 =⇒ a0 = =1 an (The equality with the exclamation point is what we want.) . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 8 / 37
17. 17. Let's be conventional The desire that these properties remain true gives us conventions for ax when x is not a positive whole number. For example, what should a0 be? We cannot write down zero a’s and multiply them together. But we would want this to be true: ! ! an an = an+0 = an · a0 =⇒ a0 = =1 an (The equality with the exclamation point is what we want.) Definition If a ̸= 0, we define a0 = 1. . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 8 / 37
18. 18. Let's be conventional The desire that these properties remain true gives us conventions for ax when x is not a positive whole number. For example, what should a0 be? We cannot write down zero a’s and multiply them together. But we would want this to be true: ! ! an an = an+0 = an · a0 =⇒ a0 = =1 an (The equality with the exclamation point is what we want.) Definition If a ̸= 0, we define a0 = 1. Notice 00 remains undefined (as a limit form, it’s indeterminate). . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 8 / 37
19. 19. Conventions for negative exponents If n ≥ 0, we want an+(−n) = an · a−n ! . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 9 / 37
20. 20. Conventions for negative exponents If n ≥ 0, we want a0 1 an+(−n) = an · a−n =⇒ a−n = ! ! n = n a a . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 9 / 37
21. 21. Conventions for negative exponents If n ≥ 0, we want a0 1 an+(−n) = an · a−n =⇒ a−n = ! ! n = n a a Definition 1 If n is a positive integer, we define a−n = . an . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 9 / 37
22. 22. Conventions for negative exponents If n ≥ 0, we want a0 1 an+(−n) = an · a−n =⇒ a−n = ! ! n = n a a Definition 1 If n is a positive integer, we define a−n = . an Fact 1 The convention that a−n = “works” for negative n as well. an am If m and n are any integers, then am−n = n . a . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 9 / 37
23. 23. Conventions for fractional exponents If q is a positive integer, we want ! (a1/q )q = a1 = a . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 10 / 37
24. 24. Conventions for fractional exponents If q is a positive integer, we want ! ! √ (a1/q )q = a1 = a =⇒ a1/q = q a . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 10 / 37
25. 25. Conventions for fractional exponents If q is a positive integer, we want ! ! √ (a1/q )q = a1 = a =⇒ a1/q = q a Definition √ If q is a positive integer, we define a1/q = q a. We must have a ≥ 0 if q is even. . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 10 / 37
26. 26. Conventions for fractional exponents If q is a positive integer, we want ! ! √ (a1/q )q = a1 = a =⇒ a1/q = q a Definition √ If q is a positive integer, we define a1/q = q a. We must have a ≥ 0 if q is even. √q ( √ )p Notice that ap = q a . So we can unambiguously say ap/q = (ap )1/q = (a1/q )p . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 10 / 37
27. 27. Conventions for irrational exponents So ax is well-defined if a is positive and x is rational. What about irrational powers? . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 11 / 37
28. 28. Conventions for irrational exponents So ax is well-defined if a is positive and x is rational. What about irrational powers? Definition Let a > 0. Then ax = lim ar r→x r rational . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 11 / 37
29. 29. Conventions for irrational exponents So ax is well-defined if a is positive and x is rational. What about irrational powers? Definition Let a > 0. Then ax = lim ar r→x r rational In other words, to approximate ax for irrational x, take r close to x but rational and compute ar . . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 11 / 37
30. 30. Approximating a power with an irrational exponent r 2r 3 23 √ =8 10 3.1 231/10 = √ 31 ≈ 8.57419 2 100 3.14 2314/100 = √2314 ≈ 8.81524 1000 3.141 23141/1000 = 23141 ≈ 8.82135 The limit (numerically approximated is) 2π ≈ 8.82498 . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 12 / 37
31. 31. Graphs of various exponential functions y . . x . . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
32. 32. Graphs of various exponential functions y . . = 1x y . x . . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
33. 33. Graphs of various exponential functions y . . = 2x y . = 1x y . x . . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
34. 34. Graphs of various exponential functions y . . = 3x. = 2x y y . = 1x y . x . . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
35. 35. Graphs of various exponential functions y . . = 10x= 3x. = 2x y y . y . = 1x y . x . . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
36. 36. Graphs of various exponential functions y . . = 10x= 3x. = 2x y y . y . = 1.5x y . = 1x y . x . . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
37. 37. Graphs of various exponential functions y . . = (1/2)x y . = 10x= 3x. = 2x y y . y . = 1.5x y . = 1x y . x . . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
38. 38. Graphs of various exponential functions x y . . = (1/2)x (1/3) y y . = . = 10x= 3x. = 2x y y . y . = 1.5x y . = 1x y . x . . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
39. 39. Graphs of various exponential functions y . . = (1/2)x (1/3) y y . = x . = (1/10)x. = 10x= 3x. = 2x y y y . y . = 1.5x y . = 1x y . x . . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
40. 40. Graphs of various exponential functions y . y yx .. = ((1/2)x (1/3)x y = 2/. )= 3 . = (1/10)x. = 10x= 3x. = 2x y y y . y . = 1.5x y . = 1x y . x . . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 37
41. 41. Outline Definition of exponential functions Properties of exponential Functions The number e and the natural exponential function Compound Interest The number e A limit Logarithmic Functions . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 14 / 37
42. 42. Properties of exponential Functions . Theorem If a > 0 and a ̸= 1, then f(x) = ax is a continuous function with domain (−∞, ∞) and range (0, ∞). In particular, ax > 0 for all x. For any real numbers x and y, and positive numbers a and b we have ax+y = ax ay ax ax−y = y a (ax )y = axy (ab)x = ax bx Proof. This is true for positive integer exponents by natural definition Our conventional definitions make these true for rational exponents Our limit definition make these for irrational exponents, too . . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 15 / 37
43. 43. Properties of exponential Functions . Theorem If a > 0 and a ̸= 1, then f(x) = ax is a continuous function with domain (−∞, ∞) and range (0, ∞). In particular, ax > 0 for all x. For any real numbers x and y, and positive numbers a and b we have ax+y = ax ay ax ax−y = y (negative exponents mean reciprocals) a (ax )y = axy (ab)x = ax bx Proof. This is true for positive integer exponents by natural definition Our conventional definitions make these true for rational exponents Our limit definition make these for irrational exponents, too . . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 15 / 37
44. 44. Properties of exponential Functions . Theorem If a > 0 and a ̸= 1, then f(x) = ax is a continuous function with domain (−∞, ∞) and range (0, ∞). In particular, ax > 0 for all x. For any real numbers x and y, and positive numbers a and b we have ax+y = ax ay ax ax−y = y (negative exponents mean reciprocals) a (ax )y = axy (fractional exponents mean roots) (ab)x = ax bx Proof. This is true for positive integer exponents by natural definition Our conventional definitions make these true for rational exponents Our limit definition make these for irrational exponents, too . . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 15 / 37
45. 45. Simplifying exponential expressions Example Simplify: 82/3 . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 16 / 37
46. 46. Simplifying exponential expressions Example Simplify: 82/3 Solution √ 3 √ 82/3 = 82 = 3 64 = 4 . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 16 / 37
47. 47. Simplifying exponential expressions Example Simplify: 82/3 Solution √3 √ 82/3 = 82 = 64 = 4 3 ( √ )2 8 = 22 = 4. 3 Or, . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 16 / 37
48. 48. Simplifying exponential expressions Example Simplify: 82/3 Solution √3 √ 82/3 = 82 = 64 = 4 3 ( √ )2 8 = 22 = 4. 3 Or, Example √ 8 Simplify: 21/2 . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 16 / 37
49. 49. Simplifying exponential expressions Example Simplify: 82/3 Solution √3 √ 82/3 = 82 = 64 = 4 3 ( √ )2 8 = 22 = 4. 3 Or, Example √ 8 Simplify: 21/2 Answer 2 . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 16 / 37
50. 50. Limits of exponential functions Fact (Limits of exponential y . functions) . = (= 2()1/32/3)x y . 1/ =x( )x y . y y y = x . 3x y . = (. /10)10x= 2x. = 1 . = y y If a > 1, then lim ax = ∞ x→∞ and lim ax = 0 x→−∞ If 0 < a < 1, then lim ax = 0 and y . = x→∞ lim a = ∞ x . x . x→−∞ . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 17 / 37
51. 51. Outline Definition of exponential functions Properties of exponential Functions The number e and the natural exponential function Compound Interest The number e A limit Logarithmic Functions . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 18 / 37
52. 52. Compounded Interest Question Suppose you save \$100 at 10% annual interest, with interest compounded once a year. How much do you have After one year? After two years? after t years? . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 19 / 37
53. 53. Compounded Interest Question Suppose you save \$100 at 10% annual interest, with interest compounded once a year. How much do you have After one year? After two years? after t years? Answer \$100 + 10% = \$110 . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 19 / 37
54. 54. Compounded Interest Question Suppose you save \$100 at 10% annual interest, with interest compounded once a year. How much do you have After one year? After two years? after t years? Answer \$100 + 10% = \$110 \$110 + 10% = \$110 + \$11 = \$121 . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 19 / 37
55. 55. Compounded Interest Question Suppose you save \$100 at 10% annual interest, with interest compounded once a year. How much do you have After one year? After two years? after t years? Answer \$100 + 10% = \$110 \$110 + 10% = \$110 + \$11 = \$121 \$100(1.1)t . . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 19 / 37
56. 56. Compounded Interest: quarterly Question Suppose you save \$100 at 10% annual interest, with interest compounded four times a year. How much do you have After one year? After two years? after t years? . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 20 / 37
57. 57. Compounded Interest: quarterly Question Suppose you save \$100 at 10% annual interest, with interest compounded four times a year. How much do you have After one year? After two years? after t years? Answer \$100(1.025)4 = \$110.38, . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 20 / 37
58. 58. Compounded Interest: quarterly Question Suppose you save \$100 at 10% annual interest, with interest compounded four times a year. How much do you have After one year? After two years? after t years? Answer \$100(1.025)4 = \$110.38, not \$100(1.1)4 ! . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 20 / 37
59. 59. Compounded Interest: quarterly Question Suppose you save \$100 at 10% annual interest, with interest compounded four times a year. How much do you have After one year? After two years? after t years? Answer \$100(1.025)4 = \$110.38, not \$100(1.1)4 ! \$100(1.025)8 = \$121.84 . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 20 / 37
60. 60. Compounded Interest: quarterly Question Suppose you save \$100 at 10% annual interest, with interest compounded four times a year. How much do you have After one year? After two years? after t years? Answer \$100(1.025)4 = \$110.38, not \$100(1.1)4 ! \$100(1.025)8 = \$121.84 \$100(1.025)4t . . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 20 / 37
61. 61. Compounded Interest: monthly Question Suppose you save \$100 at 10% annual interest, with interest compounded twelve times a year. How much do you have after t years? . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 21 / 37
62. 62. Compounded Interest: monthly Question Suppose you save \$100 at 10% annual interest, with interest compounded twelve times a year. How much do you have after t years? Answer \$100(1 + 10%/12)12t . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 21 / 37
63. 63. Compounded Interest: general Question Suppose you save P at interest rate r, with interest compounded n times a year. How much do you have after t years? . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 22 / 37
64. 64. Compounded Interest: general Question Suppose you save P at interest rate r, with interest compounded n times a year. How much do you have after t years? Answer ( r )nt B(t) = P 1 + n . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 22 / 37
65. 65. Compounded Interest: continuous Question Suppose you save P at interest rate r, with interest compounded every instant. How much do you have after t years? . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 23 / 37
66. 66. Compounded Interest: continuous Question Suppose you save P at interest rate r, with interest compounded every instant. How much do you have after t years? Answer ( ( ) r )nt 1 rnt B(t) = lim P 1 + = lim P 1 + n→∞ n n→∞ n [ ( )n ]rt 1 =P lim 1 + n→∞ n independent of P, r, or t . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 23 / 37
67. 67. The magic number Definition ( ) 1 n e = lim 1 + n→∞ n . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 24 / 37
68. 68. The magic number Definition ( ) 1 n e = lim 1 + n→∞ n So now continuously-compounded interest can be expressed as B(t) = Pert . . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 24 / 37
69. 69. Existence of e See Appendix B ( ) 1 n n 1+ n 1 2 2 2.25 . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 25 / 37
70. 70. Existence of e See Appendix B ( ) 1 n n 1+ n 1 2 2 2.25 3 2.37037 . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 25 / 37
71. 71. Existence of e See Appendix B ( ) 1 n n 1+ n 1 2 2 2.25 3 2.37037 10 2.59374 . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 25 / 37
72. 72. Existence of e See Appendix B ( ) 1 n n 1+ n 1 2 2 2.25 3 2.37037 10 2.59374 100 2.70481 . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 25 / 37
73. 73. Existence of e See Appendix B ( ) 1 n n 1+ n 1 2 2 2.25 3 2.37037 10 2.59374 100 2.70481 1000 2.71692 . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 25 / 37
74. 74. Existence of e See Appendix B ( ) 1 n n 1+ n 1 2 2 2.25 3 2.37037 10 2.59374 100 2.70481 1000 2.71692 106 2.71828 . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 25 / 37
75. 75. Existence of e See Appendix B ( ) 1 n n 1+ We can experimentally n verify that this number 1 2 exists and is 2 2.25 3 2.37037 e ≈ 2.718281828459045 . . . 10 2.59374 100 2.70481 1000 2.71692 106 2.71828 . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 25 / 37
76. 76. Existence of e See Appendix B ( ) 1 n n 1+ We can experimentally n verify that this number 1 2 exists and is 2 2.25 3 2.37037 e ≈ 2.718281828459045 . . . 10 2.59374 e is irrational 100 2.70481 1000 2.71692 106 2.71828 . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 25 / 37
77. 77. Existence of e See Appendix B ( ) 1 n n 1+ We can experimentally n verify that this number 1 2 exists and is 2 2.25 3 2.37037 e ≈ 2.718281828459045 . . . 10 2.59374 e is irrational 100 2.70481 1000 2.71692 e is transcendental 106 2.71828 . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 25 / 37
78. 78. Meet the Mathematician: Leonhard Euler Born in Switzerland, lived in Prussia (Germany) and Russia Eyesight trouble all his life, blind from 1766 onward Hundreds of contributions to calculus, number theory, graph theory, fluid mechanics, optics, and astronomy Leonhard Paul Euler Swiss, 1707–1783 . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 26 / 37
79. 79. A limit . Question eh − 1 What is lim ? h→0 h . . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 27 / 37
80. 80. A limit . Question eh − 1 What is lim ? h→0 h Answer e = lim → ∞ (1 + 1/n)n = lim (1 + h)1/h . So for a small h, e ≈ (1 + h)1/h . n h→0 So [ ]h eh − 1 (1 + h)1/h − 1 ≈ =1 h h . . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 27 / 37
81. 81. A limit . Question eh − 1 What is lim ? h→0 h Answer e = lim → ∞ (1 + 1/n)n = lim (1 + h)1/h . So for a small h, e ≈ (1 + h)1/h . n h→0 So [ ]h eh − 1 (1 + h)1/h − 1 ≈ =1 h h eh − 1 It follows that lim = 1. h→0 h 2h − 1 This can be used to characterize e: lim = 0.693 · · · < 1 and h→0 h 3h − 1 lim = 1.099 · · · > 1 h→0 h . . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 27 / 37
82. 82. Outline Definition of exponential functions Properties of exponential Functions The number e and the natural exponential function Compound Interest The number e A limit Logarithmic Functions . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 28 / 37
83. 83. Logarithms Definition The base a logarithm loga x is the inverse of the function ax y = loga x ⇐⇒ x = ay The natural logarithm ln x is the inverse of ex . So y = ln x ⇐⇒ x = ey . . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 29 / 37
84. 84. Logarithms Definition The base a logarithm loga x is the inverse of the function ax y = loga x ⇐⇒ x = ay The natural logarithm ln x is the inverse of ex . So y = ln x ⇐⇒ x = ey . Facts (i) loga (x1 · x2 ) = loga x1 + loga x2 . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 29 / 37
85. 85. Logarithms Definition The base a logarithm loga x is the inverse of the function ax y = loga x ⇐⇒ x = ay The natural logarithm ln x is the inverse of ex . So y = ln x ⇐⇒ x = ey . Facts (i) loga (x1 · x2 ) = loga x1 + loga x2 ( ) x1 (ii) loga = loga x1 − loga x2 x2 . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 29 / 37
86. 86. Logarithms Definition The base a logarithm loga x is the inverse of the function ax y = loga x ⇐⇒ x = ay The natural logarithm ln x is the inverse of ex . So y = ln x ⇐⇒ x = ey . Facts (i) loga (x1 · x2 ) = loga x1 + loga x2 ( ) x1 (ii) loga = loga x1 − loga x2 x2 (iii) loga (xr ) = r loga x . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 29 / 37
87. 87. Logarithms convert products to sums Suppose y1 = loga x1 and y2 = loga x2 Then x1 = ay1 and x2 = ay2 So x1 x2 = ay1 ay2 = ay1 +y2 Therefore loga (x1 · x2 ) = loga x1 + loga x2 . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 30 / 37
88. 88. Example Write as a single logarithm: 2 ln 4 − ln 3. . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 31 / 37
89. 89. Example Write as a single logarithm: 2 ln 4 − ln 3. Solution 42 2 ln 4 − ln 3 = ln 42 − ln 3 = ln 3 ln 42 not ! ln 3 . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 31 / 37
90. 90. Example Write as a single logarithm: 2 ln 4 − ln 3. Solution 42 2 ln 4 − ln 3 = ln 42 − ln 3 = ln 3 ln 42 not ! ln 3 Example 3 Write as a single logarithm: ln + 4 ln 2 4 . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 31 / 37
91. 91. Example Write as a single logarithm: 2 ln 4 − ln 3. Solution 42 2 ln 4 − ln 3 = ln 42 − ln 3 = ln 3 ln 42 not ! ln 3 Example 3 Write as a single logarithm: ln + 4 ln 2 4 Answer ln 12 . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 31 / 37
92. 92. Graphs of logarithmic functions y . . = 2x y y . = log2 x . . 0, 1) ( ..1, 0) . ( x . . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 32 / 37
93. 93. Graphs of logarithmic functions y . . = 3x= 2x y . y y . = log2 x y . = log3 x . . 0, 1) ( ..1, 0) . ( x . . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 32 / 37
94. 94. Graphs of logarithmic functions y . . = .10x 3x= 2x y y= . y y . = log2 x y . = log3 x . . 0, 1) ( y . = log10 x ..1, 0) . ( x . . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 32 / 37
95. 95. Graphs of logarithmic functions y . . = .10=3x= 2x y xy y y. = .ex y . = log2 x y . = ln x y . = log3 x . . 0, 1) ( y . = log10 x ..1, 0) . ( x . . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 32 / 37
96. 96. Change of base formula for exponentials Fact If a > 0 and a ̸= 1, then ln x loga x = ln a . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 33 / 37
97. 97. Change of base formula for exponentials Fact If a > 0 and a ̸= 1, then ln x loga x = ln a Proof. If y = loga x, then x = ay So ln x = ln(ay ) = y ln a Therefore ln x y = loga x = ln a . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 33 / 37
98. 98. Example of changing base Example Find log2 8 by using log10 only. Surprised? . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 34 / 37
99. 99. Example of changing base Example Find log2 8 by using log10 only. Solution log10 8 0.90309 log2 8 = ≈ =3 log10 2 0.30103 Surprised? . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 34 / 37
100. 100. Example of changing base Example Find log2 8 by using log10 only. Solution log10 8 0.90309 log2 8 = ≈ =3 log10 2 0.30103 Surprised? No, log2 8 = log2 23 = 3 directly. . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 34 / 37
101. 101. Upshot of changing base The point of the change of base formula logb x 1 loga x = = · logb x = constant · logb x logb a logb a is that all the logarithmic functions are multiples of each other. So just pick one and call it your favorite. Engineers like the common logarithm log = log10 Computer scientists like the binary logarithm lg = log2 Mathematicians like natural logarithm ln = loge . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 35 / 37
102. 102. “ . . lawn” . . . . . . . . Image credit: Selva V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 36 / 37
103. 103. Summary Exponentials turn sums into products Logarithms turn products into sums Slide rule scabbards are wicked cool . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 37 / 37