1. Sections 3.1–3.2 Exponential and Logarithmic Functions V63.0121.041, Calculus I New York University October 20, 2010Announcements Midterm is graded and scores are on blackboard. Should get it back in recitation. There is WebAssign due Monday/Tuesday next week. . . . . . .
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. 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. OutlineDefinition of exponential functionsProperties of exponential FunctionsThe number e and the natural exponential function Compound Interest The number e A limitLogarithmic Functions . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 4 / 37
5. Derivation of exponential functionsDefinitionIf 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. Derivation of exponential functionsDefinitionIf a is a real number and n is a positive whole number, then an = a · a · · · · · a n factorsExamples 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. Anatomy of a powerDefinitionA 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. FactIf a is a real number, then ax+y = ax ay (sums to products) ax ax−y = y a (ax )y = axy (ab)x = ax bxwhenever 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. FactIf 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 bxwhenever 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. FactIf 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 bxwhenever 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. FactIf 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. FactIf 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. FactIf 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. Lets 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. Lets 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. Lets 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. Lets 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.)DefinitionIf 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. Lets 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.)DefinitionIf 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. Conventions for negative exponentsIf 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. Conventions for negative exponentsIf 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. Conventions for negative exponentsIf n ≥ 0, we want a0 1 an+(−n) = an · a−n =⇒ a−n = ! ! n = n a aDefinition 1If 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. Conventions for negative exponentsIf n ≥ 0, we want a0 1 an+(−n) = an · a−n =⇒ a−n = ! ! n = n a aDefinition 1If n is a positive integer, we define a−n = . anFact 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. Conventions for fractional exponentsIf 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. Conventions for fractional exponentsIf 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. Conventions for fractional exponentsIf q is a positive integer, we want ! ! √ (a1/q )q = a1 = a =⇒ a1/q = q aDefinition √If q is a positive integer, we define a1/q = q a. We must have a ≥ 0 if qis even. . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 10 / 37
26. Conventions for fractional exponentsIf q is a positive integer, we want ! ! √ (a1/q )q = a1 = a =⇒ a1/q = q aDefinition √If q is a positive integer, we define a1/q = q a. We must have a ≥ 0 if qis even. √q ( √ )pNotice 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. 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. Conventions for irrational exponents So ax is well-defined if a is positive and x is rational. What about irrational powers?DefinitionLet 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. Conventions for irrational exponents So ax is well-defined if a is positive and x is rational. What about irrational powers?DefinitionLet a > 0. Then ax = lim ar r→x r rationalIn other words, to approximate ax for irrational x, take r close to x butrational and compute ar . . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 11 / 37
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 ≈ 8.82135The 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. OutlineDefinition of exponential functionsProperties of exponential FunctionsThe number e and the natural exponential function Compound Interest The number e A limitLogarithmic Functions . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 14 / 37
42. Properties of exponential Functions.TheoremIf 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 numbersx 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 bxProof. 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. Properties of exponential Functions.TheoremIf 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 numbersx 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 bxProof. 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. Properties of exponential Functions.TheoremIf 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 numbersx 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 bxProof. 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
50. Limits of exponential functionsFact (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. OutlineDefinition of exponential functionsProperties of exponential FunctionsThe number e and the natural exponential function Compound Interest The number e A limitLogarithmic Functions . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 18 / 37
52. Compounded InterestQuestionSuppose you save $100 at 10% annual interest, with interestcompounded 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. Compounded InterestQuestionSuppose you save $100 at 10% annual interest, with interestcompounded 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. Compounded InterestQuestionSuppose you save $100 at 10% annual interest, with interestcompounded 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. Compounded InterestQuestionSuppose you save $100 at 10% annual interest, with interestcompounded 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. Compounded Interest: quarterlyQuestionSuppose you save $100 at 10% annual interest, with interestcompounded 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. Compounded Interest: quarterlyQuestionSuppose you save $100 at 10% annual interest, with interestcompounded 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. Compounded Interest: quarterlyQuestionSuppose you save $100 at 10% annual interest, with interestcompounded 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. Compounded Interest: quarterlyQuestionSuppose you save $100 at 10% annual interest, with interestcompounded 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. Compounded Interest: quarterlyQuestionSuppose you save $100 at 10% annual interest, with interestcompounded 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. Compounded Interest: monthlyQuestionSuppose you save $100 at 10% annual interest, with interestcompounded twelve times a year. How much do you have after tyears? . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 21 / 37
62. Compounded Interest: monthlyQuestionSuppose you save $100 at 10% annual interest, with interestcompounded twelve times a year. How much do you have after tyears?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. Compounded Interest: generalQuestionSuppose you save P at interest rate r, with interest compounded ntimes 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. Compounded Interest: generalQuestionSuppose you save P at interest rate r, with interest compounded ntimes 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. Compounded Interest: continuousQuestionSuppose you save P at interest rate r, with interest compounded everyinstant. 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. Compounded Interest: continuousQuestionSuppose you save P at interest rate r, with interest compounded everyinstant. 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. The magic numberDefinition ( ) 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. The magic numberDefinition ( ) 1 n e = lim 1 + n→∞ nSo 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. Existence of eSee 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. Existence of eSee 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. Existence of eSee 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. Existence of eSee 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. Existence of eSee 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. Existence of eSee 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. Existence of eSee 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. Existence of eSee 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. Existence of eSee 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. 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. A limit.Question eh − 1What is lim ? h→0 h. . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 27 / 37
80. A limit.Question eh − 1What is lim ? h→0 hAnswer 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. A limit.Question eh − 1What is lim ? h→0 hAnswer 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. OutlineDefinition of exponential functionsProperties of exponential FunctionsThe number e and the natural exponential function Compound Interest The number e A limitLogarithmic Functions . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 28 / 37
83. LogarithmsDefinition 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. LogarithmsDefinition 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. LogarithmsDefinition 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. LogarithmsDefinition 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. 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. ExampleWrite 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. ExampleWrite 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. ExampleWrite 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 3Example 3Write 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. ExampleWrite 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 3Example 3Write as a single logarithm: ln + 4 ln 2 4Answerln 12 . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 31 / 37
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. 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. 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. 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. Change of base formula for exponentialsFactIf 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. Change of base formula for exponentialsFactIf a > 0 and a ̸= 1, then ln x loga x = ln aProof. 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. Example of changing baseExampleFind 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. Example of changing baseExampleFind log2 8 by using log10 only.Solution log10 8 0.90309log2 8 = ≈ =3 log10 2 0.30103Surprised? . . . . . . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 34 / 37
100. Example of changing baseExampleFind log2 8 by using log10 only.Solution log10 8 0.90309log2 8 = ≈ =3 log10 2 0.30103Surprised? 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. Upshot of changing baseThe point of the change of base formula logb x 1 loga x = = · logb x = constant · logb x logb a logb ais that all the logarithmic functions are multiples of each other. So justpick 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
Be the first to comment