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2.4 introduction to logarithm

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• 1. The Logarithmic Functions
• 2. The Logarithmic FunctionsThe identity expression 82 = 64 has three parts.
• 3. The Logarithmic FunctionsThe identity expression 82 = 64 has three parts. 82 = 64
• 4. The Logarithmic FunctionsThe identity expression 82 = 64 has three parts. the base 82 = 64
• 5. The Logarithmic FunctionsThe identity expression 82 = 64 has three parts. the exponent the base 82 = 64
• 6. The Logarithmic FunctionsThe identity expression 82 = 64 has three parts. the base 82 = 64 the outcome
• 7. The Logarithmic FunctionsThe identity expression 82 = 64 has three parts. the exponent the base 82 = 64 the outcome
• 8. The Logarithmic FunctionsThe identity expression 82 = 64 has three parts. the exponent the base 82 = 64 the outcomeThe exponent 2 has another name which indicates itsrole in the expression.
• 9. The Logarithmic FunctionsThe identity expression 82 = 64 has three parts. the exponent the base 82 = 64 the outcomeThe exponent 2 has another name which indicates itsrole in the expression. Specifically, 2 is the exponentthat will give the outcome 64.
• 10. The Logarithmic FunctionsThe identity expression 82 = 64 has three parts. the exponent the base 82 = 64 the outcomeThe exponent 2 has another name which indicates itsrole in the expression. Specifically, 2 is the exponentthat will give the outcome 64. We write this as 2 = log (64)
• 11. The Logarithmic FunctionsThe identity expression 82 = 64 has three parts. the exponent the base 82 = 64 the outcomeThe exponent 2 has another name which indicates itsrole in the expression. Specifically, 2 is the exponentthat will give the outcome 64. We write this as 2 = log (64) the exponent
• 12. The Logarithmic FunctionsThe identity expression 82 = 64 has three parts. the exponent the base 82 = 64 the outcomeThe exponent 2 has another name which indicates itsrole in the expression. Specifically, 2 is the exponentthat will give the outcome 64. We write this as 2 = log8(64) the exponent size of the base
• 13. The Logarithmic FunctionsThe identity expression 82 = 64 has three parts. the exponent the base 82 = 64 the outcomeThe exponent 2 has another name which indicates itsrole in the expression. Specifically, 2 is the exponentthat will give the outcome 64. We write this as 2 = log8(64) the exponent size of the baseSimilarly the expression 43 = 64 may be rephrased as 3 = log (64)
• 14. The Logarithmic FunctionsThe identity expression 82 = 64 has three parts. the exponent the base 82 = 64 the outcomeThe exponent 2 has another name which indicates itsrole in the expression. Specifically, 2 is the exponentthat will give the outcome 64. We write this as 2 = log8(64) the exponent size of the baseSimilarly the expression 43 = 64 may be rephrased as 3 = log (64) the exponent
• 15. The Logarithmic FunctionsThe identity expression 82 = 64 has three parts. the exponent the base 82 = 64 the outcomeThe exponent 2 has another name which indicates itsrole in the expression. Specifically, 2 is the exponentthat will give the outcome 64. We write this as 2 = log8(64) the exponent size of the baseSimilarly the expression 43 = 64 may be rephrased as 3 = log4(64) the exponent size of the base
• 16. The Logarithmic FunctionsHence logb(y) = the exponent x, where x is thepower needed (with base b) to get y.
• 17. The Logarithmic FunctionsHence logb(y) = the exponent x, where x is thepower needed (with base b) to get y.The expressions bx = y and x = logb(y) describe thesame relation between the numbers b, x and y.
• 18. The Logarithmic FunctionsHence logb(y) = the exponent x, where x is thepower needed (with base b) to get y.The expressions bx = y and x = logb(y) describe thesame relation between the numbers b, x and y.The expression bx = y is called the exp-form.
• 19. The Logarithmic FunctionsHence logb(y) = the exponent x, where x is thepower needed (with base b) to get y.The expressions bx = y and x = logb(y) describe thesame relation between the numbers b, x and y.The expression bx = y is called the exp-form.The form logb(y) = x is called the log-form.
• 20. The Logarithmic FunctionsHence logb(y) = the exponent x, where x is thepower needed (with base b) to get y.The expressions bx = y and x = logb(y) describe thesame relation between the numbers b, x and y.The expression bx = y is called the exp-form.The form logb(y) = x is called the log-form.Example A. Rewrite the exp-form into the log-form.a. 42 = 16b. w = u2+v
• 21. The Logarithmic FunctionsHence logb(y) = the exponent x, where x is thepower needed (with base b) to get y.The expressions bx = y and x = logb(y) describe thesame relation between the numbers b, x and y.The expression bx = y is called the exp-form.The form logb(y) = x is called the log-form.Example A. Rewrite the exp-form into the log-form.a. 42 = 16  2 = log4(16)b. w = u2+v
• 22. The Logarithmic FunctionsHence logb(y) = the exponent x, where x is thepower needed (with base b) to get y.The expressions bx = y and x = logb(y) describe thesame relation between the numbers b, x and y.The expression bx = y is called the exp-form.The form logb(y) = x is called the log-form.Example A. Rewrite the exp-form into the log-form.a. 42 = 16  2 = log4(16)b. w = u2+v  logu(w) = 2+v
• 23. The Logarithmic FunctionsHence logb(y) = the exponent x, where x is thepower needed (with base b) to get y.The expressions bx = y and x = logb(y) describe thesame relation between the numbers b, x and y.The expression bx = y is called the exp-form.The form logb(y) = x is called the log-form.Example A. Rewrite the exp-form into the log-form.a. 42 = 16  2 = log4(16)b. w = u2+v  logu(w) = 2+vExample B. Rewrite the log-form into the exp-form.a. log3(1/9) = –2b. 2w = logv(a – b)
• 24. The Logarithmic FunctionsHence logb(y) = the exponent x, where x is thepower needed (with base b) to get y.The expressions bx = y and x = logb(y) describe thesame relation between the numbers b, x and y.The expression bx = y is called the exp-form.The form logb(y) = x is called the log-form.Example A. Rewrite the exp-form into the log-form.a. 42 = 16  2 = log4(16)b. w = u2+v  logu(w) = 2+vExample B. Rewrite the log-form into the exp-form.a. log3(1/9) = –2  1/9 = 3–2b. 2w = logv(a – b)
• 25. The Logarithmic FunctionsHence logb(y) = the exponent x, where x is thepower needed (with base b) to get y.The expressions bx = y and x = logb(y) describe thesame relation between the numbers b, x and y.The expression bx = y is called the exp-form.The form logb(y) = x is called the log-form.Example A. Rewrite the exp-form into the log-form.a. 42 = 16  2 = log4(16)b. w = u2+v  logu(w) = 2+vExample B. Rewrite the log-form into the exp-form.a. log3(1/9) = –2  1/9 = 3–2b. 2w = logv(a – b)  a – b = v2w
• 26. The Logarithmic FunctionsSince bx is positive for all xs, so the domain of logb(y)is the set of all y > 0.
• 27. The Logarithmic FunctionsSince bx is positive for all xs, so the domain of logb(y)is the set of all y > 0.For example, log2(–1) doesnt exist because there isno exponent x such that 2x = –1.
• 28. The Logarithmic FunctionsSince bx is positive for all xs, so the domain of logb(y)is the set of all y > 0.For example, log2(–1) doesnt exist because there isno exponent x such that 2x = –1. Take out your calculator enter log (–1) and see what it returns.
• 29. The Logarithmic FunctionsSince bx is positive for all xs, so the domain of logb(y)is the set of all y > 0.For example, log2(–1) doesnt exist because there isno exponent x such that 2x = –1. Graphs of the Logarithmic Functions
• 30. The Logarithmic FunctionsSince bx is positive for all xs, so the domain of logb(y)is the set of all y > 0.For example, log2(–1) doesnt exist because there isno exponent x such that 2x = –1. Graphs of the Logarithmic Functions x y=log2(x) 1/4 1/2 1 2 4 8
• 31. The Logarithmic FunctionsSince bx is positive for all xs, so the domain of logb(y)is the set of all y > 0.For example, log2(–1) doesnt exist because there isno exponent x such that 2x = –1. Graphs of the Logarithmic Functions x y=log2(x) 1/4 -2 1/2 1 2 4 8
• 32. The Logarithmic FunctionsSince bx is positive for all xs, so the domain of logb(y)is the set of all y > 0.For example, log2(–1) doesnt exist because there isno exponent x such that 2x = –1. Graphs of the Logarithmic Functions x y=log2(x) 1/4 -2 1/2 -1 1 2 4 8
• 33. The Logarithmic FunctionsSince bx is positive for all xs, so the domain of logb(y)is the set of all y > 0.For example, log2(–1) doesnt exist because there isno exponent x such that 2x = –1. Graphs of the Logarithmic Functions x y=log2(x) 1/4 -2 1/2 -1 1 0 2 4 8
• 34. The Logarithmic FunctionsSince bx is positive for all xs, so the domain of logb(y)is the set of all y > 0.For example, log2(–1) doesnt exist because there isno exponent x such that 2x = –1. Graphs of the Logarithmic Functions x y=log2(x) 1/4 -2 1/2 -1 1 0 2 1 4 8
• 35. The Logarithmic FunctionsSince bx is positive for all xs, so the domain of logb(y)is the set of all y > 0.For example, log2(–1) doesnt exist because there isno exponent x such that 2x = –1. Graphs of the Logarithmic Functions x y=log2(x) 1/4 -2 1/2 -1 1 0 2 1 4 2 8 3
• 36. The Logarithmic FunctionsSince bx is positive for all xs, so the domain of logb(y)is the set of all y > 0.For example, log2(–1) doesnt exist because there isno exponent x such that 2x = –1. Graphs of the Logarithmic Functions x y=log2(x) 1/4 -2 (16, 4) (8, 3) 1/2 -1 (4, 2) (2, 1) 1 0 (1, 0) 2 1 (1/2, -1) 4 2 (1/4, -2) y=log2(x) 8 3
• 37. The Logarithmic Functionsy = logb(x) for b > 1. y x (1, 0)
• 38. The Logarithmic Functionsy = logb(x) for b > 1. y = logb(x) for 1 > b > 0. y y x x (1, 0) (1, 0)
• 39. The Logarithmic Functionsy = logb(x) for b > 1. y = logb(x) for 1 > b > 0. y y x x (1, 0) (1, 0) The Common Log and the Natural Log
• 40. The Logarithmic Functionsy = logb(x) for b > 1. y = logb(x) for 1 > b > 0. y y x x (1, 0) (1, 0) The Common Log and the Natural LogThe most often used bases are 10 and e.
• 41. The Logarithmic Functionsy = logb(x) for b > 1. y = logb(x) for 1 > b > 0. y y x x (1, 0) (1, 0) The Common Log and the Natural LogThe most often used bases are 10 and e.Base 10 is called the common base.
• 42. The Logarithmic Functionsy = logb(x) for b > 1. y = logb(x) for 1 > b > 0. y y x x (1, 0) (1, 0) The Common Log and the Natural LogThe most often used bases are 10 and e.Base 10 is called the common base.Log with base10 is called the common log.
• 43. The Logarithmic Functionsy = logb(x) for b > 1. y = logb(x) for 1 > b > 0. y y x x (1, 0) (1, 0) The Common Log and the Natural LogThe most often used bases are 10 and e.Base 10 is called the common base.Log with base10 is called the common log.It is written as log(x) without the base number b.
• 44. The Common Log and the Natural LogBase e is called the natural base.
• 45. The Common Log and the Natural LogBase e is called the natural base.Log with base e is called the natural log.
• 46. The Common Log and the Natural LogBase e is called the natural base.Log with base e is called the natural log.It is written as ln(x).
• 47. The Common Log and the Natural LogBase e is called the natural base.Log with base e is called the natural log.It is written as ln(x).Example C. Convert to the other form. exp-form log-form 103 = 1000 ln(1/e2) = -2 e rt = A P log(1) = 0
• 48. The Common Log and the Natural LogBase e is called the natural base.Log with base e is called the natural log.It is written as ln(x).Example C. Convert to the other form. exp-form log-form 103 = 1000 log(1000) = 3 ln(1/e2) = -2 e rt = A P log(1) = 0
• 49. The Common Log and the Natural LogBase e is called the natural base.Log with base e is called the natural log.It is written as ln(x).Example C. Convert to the other form. exp-form log-form 103 = 1000 log(1000) = 3 e-2 = 1/e2 ln(1/e2) = -2 e rt = A P log(1) = 0
• 50. The Common Log and the Natural LogBase e is called the natural base.Log with base e is called the natural log.It is written as ln(x).Example C. Convert to the other form. exp-form log-form 103 = 1000 log(1000) = 3 e-2 = 1/e2 ln(1/e2) = -2 e rt = A ln( A ) = rt P P log(1) = 0
• 51. The Common Log and the Natural LogBase e is called the natural base.Log with base e is called the natural log.It is written as ln(x).Example C. Convert to the other form. exp-form log-form 103 = 1000 log(1000) = 3 e-2 = 1/e2 ln(1/e2) = -2 e rt = A ln( A ) = rt P P 100 = 1 log(1) = 0
• 52. The Common Log and the Natural LogWhen we change the log-form into the exp-form,we say we "drop the log".
• 53. The Common Log and the Natural LogWhen we change the log-form into the exp-form,we say we "drop the log".Example D. Solve for xa. log9(x) = -1b. logx(9) = -2
• 54. The Common Log and the Natural LogWhen we change the log-form into the exp-form,we say we "drop the log".Example D. Solve for xa. log9(x) = -1Drop the log and get x = 9-1.b. logx(9) = -2
• 55. The Common Log and the Natural LogWhen we change the log-form into the exp-form,we say we "drop the log".Example D. Solve for xa. log9(x) = -1Drop the log and get x = 9-1. So x = 1/9b. logx(9) = -2
• 56. The Common Log and the Natural LogWhen we change the log-form into the exp-form,we say we "drop the log".Example D. Solve for xa. log9(x) = -1Drop the log and get x = 9-1. So x = 1/9b. logx(9) = -2 1Drop the log and get 9 = x-2, i.e. 9 = 2 x
• 57. The Common Log and the Natural LogWhen we change the log-form into the exp-form,we say we "drop the log".Example D. Solve for xa. log9(x) = -1Drop the log and get x = 9-1. So x = 1/9b. logx(9) = -2 1Drop the log and get 9 = x-2, i.e. 9 = 2 2 = 1 xSo 9x
• 58. The Common Log and the Natural LogWhen we change the log-form into the exp-form,we say we "drop the log".Example D. Solve for xa. log9(x) = -1Drop the log and get x = 9-1. So x = 1/9b. logx(9) = -2 1Drop the log and get 9 = x-2, i.e. 9 = 2 2 = 1 xSo 9x x2 = 1/9 x=1/3 or x=-1/3
• 59. The Common Log and the Natural LogWhen we change the log-form into the exp-form,we say we "drop the log".Example D. Solve for xa. log9(x) = -1Drop the log and get x = 9-1. So x = 1/9b. logx(9) = -2 1Drop the log and get 9 = x-2, i.e. 9 = 2 2 = 1 xSo 9x x2 = 1/9 x=1/3 or x=-1/3But the base has to be positive, hence x = 1/3 is theonly solution.
• 60. The Logarithmic FunctionsExercise A. Rewrite the following exp-form into the log-form. 21. 5 = 25 2. 1/25 = 5–2 3. 33 = 274. 1/25 = 5–2 5. 1/27 = 3–3 6. x3 = y7. y3 = x 8. 1/a = b–2 9. ep = a + b10. e(a + b) = p 11. A = e–rt 12. 10x–y = zExercise B. Rewrite the following log–form into the exp-form.13. log3(1/9) = –2 14. –2 = log4(1/16) 15. log1/3(9) = –216. log1/4(16) = –2 17. 2w = logv(a – b) 18. logv(2w) = a – b19. log (1/100) = –2 20. 1/2 = log(√10) 21. ln(1/e2) = –222. log (A/B) = 3 23. rt = ln(ert) 24. ln(1/√e) = –1/2
• 61. The Logarithmic FunctionsExercise C. Convert the following into the exponential formthen solve for x.25. logx(9) = 2 26. x = log2(8) 27. log3(x) = 228. logx(x) = 2 29. 2 = log2(x) 30. logx(x + 2) = 231. log1/2(4) = x 32. 4 = log1/2(x) 33. logx(4) = 1/224. ln(x) = 2 35. 2 = log(x) 36. log(4x + 15) = 237. In(x) = –1/2 38. a = In(2x – 3) 39. log(x2 – 15x) = 2