4.4the logarithm functions
- 2. There are three numbers in an exponential notation. The Logarithmic Functions 4 3 = 64
- 3. There are three numbers in an exponential notation. The Logarithmic Functions the base 4 3 = 64
- 4. There are three numbers in an exponential notation. The Logarithmic Functions the exponent the base 4 3 = 64
- 5. There are three numbers in an exponential notation. The Logarithmic Functions the exponent the base the output 4 3 = 64
- 6. There are three numbers in an exponential notation. Given the above expression, we say that “(base) 4 raised to the exponent (power) 3 gives 64”. The Logarithmic Functions the exponent the base the output 4 3 = 64
- 7. There are three numbers in an exponential notation. The Logarithmic Functions the exponent the base the output 4 3 = 64 Given the above expression, we say that “(base) 4 raised to the exponent (power) 3 gives 64”. The focus of the above statement is that when 43 is executed, the output is 64.
- 8. There are three numbers in an exponential notation. The Logarithmic Functions the exponent the base the output 4 3 = 64 However if we are given the output is 64 from raising 4 to a power, Given the above expression, we say that “(base) 4 raised to the exponent (power) 3 gives 64”. The focus of the above statement is that when 43 is executed, the output is 64.
- 9. There are three numbers in an exponential notation. The Logarithmic Functions the exponent the base the output 4 3 = 64 However if we are given the output is 64 from raising 4 to a power, the power the base the output 4 = 64 3 Given the above expression, we say that “(base) 4 raised to the exponent (power) 3 gives 64”. The focus of the above statement is that when 43 is executed, the output is 64.
- 10. There are three numbers in an exponential notation. The Logarithmic Functions the exponent the base the output 4 3 = 64 However if we are given the output is 64 from raising 4 to a power, then the needed power is called log4(64) the power = log4(64) the base the output 4 = 64 3 Given the above expression, we say that “(base) 4 raised to the exponent (power) 3 gives 64”. The focus of the above statement is that when 43 is executed, the output is 64.
- 11. There are three numbers in an exponential notation. The Logarithmic Functions the exponent the base the output 4 3 = 64 However if we are given the output is 64 from raising 4 to a power, then the needed power is called log4(64) which is 3. the power = log4(64) the base the output 4 = 64 3 Given the above expression, we say that “(base) 4 raised to the exponent (power) 3 gives 64”. The focus of the above statement is that when 43 is executed, the output is 64.
- 12. There are three numbers in an exponential notation. The Logarithmic Functions the exponent the base the output 4 3 = 64 However if we are given the output is 64 from raising 4 to a power, then the needed power is called log4(64) which is 3. the power = log4(64) the base the output 4 = 64 3 or that log4(64) = 3 and we say that “log–base–4 of 64 is 3”. Given the above expression, we say that “(base) 4 raised to the exponent (power) 3 gives 64”. The focus of the above statement is that when 43 is executed, the output is 64.
- 13. The Logarithmic Functions Just as the sentence “Bart's dad is Homer.” contains the same information as “Homer's son is Bart.”,
- 14. The Logarithmic Functions Just as the sentence “Bart's dad is Homer.” contains the same information as “Homer's son is Bart.” The expression “64 = 43” contains the same information as “log4(64) = 3”.
- 15. The Logarithmic Functions Just as the sentence “Bart's dad is Homer.” contains the same information as “Homer's son is Bart.” The expression “64 = 43” contains the same information as “log4(64) = 3”. The expression “64 = 43” is called the exponential form and “log4(64) = 3” is called the logarithmic form of the expressed relation.
- 16. The Logarithmic Functions Just as the sentence “Bart's dad is Homer.” contains the same information as “Homer's son is Bart.” The expression “64 = 43” contains the same information as “log4(64) = 3”. The expression “64 = 43” is called the exponential form and “log4(64) = 3” is called the logarithmic form of the expressed relation. In general, we say that “log–base–b of y is x” or logb(y) = x
- 17. The Logarithmic Functions Just as the sentence “Bart's dad is Homer.” contains the same information as “Homer's son is Bart.” The expression “64 = 43” contains the same information as “log4(64) = 3”. The expression “64 = 43” is called the exponential form and “log4(64) = 3” is called the logarithmic form of the expressed relation. In general, we say that “log–base–b of y is x” or logb(y) = x if y = bx (b > 0).
- 18. The Logarithmic Functions Just as the sentence “Bart's dad is Homer.” contains the same information as “Homer's son is Bart.” The expression “64 = 43” contains the same information as “log4(64) = 3”. The expression “64 = 43” is called the exponential form and “log4(64) = 3” is called the logarithmic form of the expressed relation. In general, we say that “log–base–b of y is x” or logb(y) = x if y = bx (b > 0). the power = logb(y) the base the output b = y x
- 19. The Logarithmic Functions Just as the sentence “Bart's dad is Homer.” contains the same information as “Homer's son is Bart.” The expression “64 = 43” contains the same information as “log4(64) = 3”. The expression “64 = 43” is called the exponential form and “log4(64) = 3” is called the logarithmic form of the expressed relation. In general, we say that “log–base–b of y is x” or logb(y) = x if y = bx (b > 0), i.e. logb(y) is the exponent x. the power = logb(y) the base the output b = y x
- 20. The Logarithmic Functions When working with the exponential form or the logarithmic expressions, always identify the base number b first.
- 21. The Logarithmic Functions When working with the exponential form or the logarithmic expressions, always identify the base number b first. All the following exponential expressions yield 64.
- 22. The Logarithmic Functions When working with the exponential form or the logarithmic expressions, always identify the base number b first. All the following exponential expressions yield 64. 43 → 64 82 → 64 26 → 64 exp–form
- 23. The Logarithmic Functions When working with the exponential form or the logarithmic expressions, always identify the base number b first. All the following exponential expressions yield 64. Their corresponding log–form are differentiated by the bases and the different exponents required. 43 → 64 82 → 64 26 → 64 exp–form log–form
- 24. The Logarithmic Functions When working with the exponential form or the logarithmic expressions, always identify the base number b first. All the following exponential expressions yield 64. 43 → 64 82 → 64 26 → 64 log4(64) log8(64) log2(64) exp–form log–formTheir corresponding log–form are differentiated by the bases and the different exponents required.
- 25. The Logarithmic Functions When working with the exponential form or the logarithmic expressions, always identify the base number b first. All the following exponential expressions yield 64. Their corresponding log–form are differentiated by the bases and the different exponents required. 43 → 64 82 → 64 26 → 64 log4(64) → log8(64) → log2(64) → exp–form log–form
- 26. The Logarithmic Functions When working with the exponential form or the logarithmic expressions, always identify the base number b first. All the following exponential expressions yield 64. 43 → 64 82 → 64 26 → 64 log4(64) → 3 log8(64) → log2(64) → exp–form log–formTheir corresponding log–form are differentiated by the bases and the different exponents required.
- 27. The Logarithmic Functions When working with the exponential form or the logarithmic expressions, always identify the base number b first. All the following exponential expressions yield 64. 43 → 64 82 → 64 26 → 64 log4(64) → 3 log8(64) → 2 log2(64) → exp–form log–formTheir corresponding log–form are differentiated by the bases and the different exponents required.
- 28. The Logarithmic Functions When working with the exponential form or the logarithmic expressions, always identify the base number b first. All the following exponential expressions yield 64. 43 → 64 82 → 64 26 → 64 log4(64) → 3 log8(64) → 2 log2(64) → 6 exp–form log–formTheir corresponding log–form are differentiated by the bases and the different exponents required.
- 29. The Logarithmic Functions When working with the exponential form or the logarithmic expressions, always identify the base number b first. All the following exponential expressions yield 64. 43 → 64 82 → 64 26 → 64 log4(64) → 3 log8(64) → 2 log2(64) → 6 exp–form log–formTheir corresponding log–form are differentiated by the bases and the different exponents required. Both numbers b and y appeared in the log notation “logb(y)” must be positive.
- 30. The Logarithmic Functions When working with the exponential form or the logarithmic expressions, always identify the base number b first. All the following exponential expressions yield 64. 43 → 64 82 → 64 26 → 64 log4(64) → 3 log8(64) → 2 log2(64) → 6 exp–form log–formTheir corresponding log–form are differentiated by the bases and the different exponents required. Both numbers b and y appeared in the log notation “logb(y)” must be positive. Switch to x as the input, the domain of logb(x) is the set D = {x l x > 0 }.
- 31. The Logarithmic Functions When working with the exponential form or the logarithmic expressions, always identify the base number b first. All the following exponential expressions yield 64. 43 → 64 82 → 64 26 → 64 log4(64) → 3 log8(64) → 2 log2(64) → 6 exp–form log–formTheir corresponding log–form are differentiated by the bases and the different exponents required. Both numbers b and y appeared in the log notation “logb(y)” must be positive. Switch to x as the input, the domain of logb(x) is the set D = {x l x > 0 }. We would get an error message if we execute log2(–1) with software.
- 32. The Logarithmic Functions To convert the exp-form to the log–form: b = y x
- 33. The Logarithmic Functions To convert the exp-form to the log–form: b = y x logb( y ) = x→ Identity the base and the correct log–function
- 34. The Logarithmic Functions To convert the exp-form to the log–form: b = y x logb( y ) = x→ insert the exponential output.
- 35. The Logarithmic Functions To convert the exp-form to the log–form: b = y x logb( y ) = x→ The log–output is the required exponent.
- 36. The Logarithmic Functions Example A. Rewrite the exp-form into the log-form. a. 42 = 16 b. w = u2+v To convert the exp-form to the log–form: b = y x logb( y ) = x→
- 37. The Logarithmic Functions Example A. Rewrite the exp-form into the log-form. a. 42 = 16 log4(16) = 2 b. w = u2+v To convert the exp-form to the log–form: b = y x logb( y ) = x→
- 38. The Logarithmic Functions Example A. Rewrite the exp-form into the log-form. a. 42 = 16 log4(16) = 2 b. w = u2+v To convert the exp-form to the log–form: b = y x logb( y ) = x→
- 39. The Logarithmic Functions Example A. Rewrite the exp-form into the log-form. a. 42 = 16 log4(16) = 2 b. w = u2+v To convert the exp-form to the log–form: b = y x logb( y ) = x→
- 40. The Logarithmic Functions Example A. Rewrite the exp-form into the log-form. a. 42 = 16 log4(16) = 2 b. w = u2+v logu(w) = 2+v To convert the exp-form to the log–form: b = y x logb( y ) = x→
- 41. The Logarithmic Functions Example A. Rewrite the exp-form into the log-form. a. 42 = 16 log4(16) = 2 b. w = u2+v logu(w) = 2+v To convert the exp-form to the log–form: b = y x logb( y ) = x→
- 42. The Logarithmic Functions Example A. Rewrite the exp-form into the log-form. a. 42 = 16 log4(16) = 2 b. w = u2+v logu(w) = 2+v To convert the exp-form to the log–form: b = y x logb( y ) = x→
- 43. The Logarithmic Functions Example A. Rewrite the exp-form into the log-form. a. 42 = 16 log4(16) = 2 b. w = u2+v logu(w) = 2+v To convert the exp-form to the log–form: b = y x → To convert the log–form to the exp–form: logb( y ) = x logb( y ) = x
- 44. The Logarithmic Functions Example A. Rewrite the exp-form into the log-form. a. 42 = 16 log4(16) = 2 b. w = u2+v logu(w) = 2+v To convert the exp-form to the log–form: b = y x → To convert the log–form to the exp–form: b = y x logb( y ) = x→ logb( y ) = x
- 45. The Logarithmic Functions Example A. Rewrite the exp-form into the log-form. a. 42 = 16 log4(16) = 2 b. w = u2+v logu(w) = 2+v To convert the exp-form to the log–form: b = y x → To convert the log–form to the exp–form: b = y x logb( y ) = x→ logb( y ) = x
- 46. The Logarithmic Functions Example A. Rewrite the exp-form into the log-form. a. 42 = 16 log4(16) = 2 b. w = u2+v logu(w) = 2+v To convert the exp-form to the log–form: b = y x → To convert the log–form to the exp–form: b = y x logb( y ) = x→ logb( y ) = x
- 47. The Logarithmic Functions Example B. Rewrite the log-form into the exp-form. a. log3(1/9) = –2 b. 2w = logv(a – b) Example A. Rewrite the exp-form into the log-form. a. 42 = 16 log4(16) = 2 b. w = u2+v logu(w) = 2+v To convert the exp-form to the log–form: b = y x logb( y ) = x→ To convert the log–form to the exp–form: b = y x logb( y ) = x→
- 48. The Logarithmic Functions Example B. Rewrite the log-form into the exp-form. a. log3(1/9) = –2 3-2 = 1/9 b. 2w = logv(a – b) Example A. Rewrite the exp-form into the log-form. a. 42 = 16 log4(16) = 2 b. w = u2+v logu(w) = 2+v To convert the exp-form to the log–form: b = y x logb( y ) = x→ To convert the log–form to the exp–form: b = y x logb( y ) = x→
- 49. The Logarithmic Functions Example B. Rewrite the log-form into the exp-form. a. log3(1/9) = –2 3-2 = 1/9 b. 2w = logv(a – b) Example A. Rewrite the exp-form into the log-form. a. 42 = 16 log4(16) = 2 b. w = u2+v logu(w) = 2+v To convert the exp-form to the log–form: b = y x logb( y ) = x→ To convert the log–form to the exp–form: b = y x logb( y ) = x→
- 50. The Logarithmic Functions Example B. Rewrite the log-form into the exp-form. a. log3(1/9) = –2 3-2 = 1/9 b. 2w = logv(a – b) Example A. Rewrite the exp-form into the log-form. a. 42 = 16 log4(16) = 2 b. w = u2+v logu(w) = 2+v To convert the exp-form to the log–form: b = y x logb( y ) = x→ To convert the log–form to the exp–form: b = y x logb( y ) = x→
- 51. The Logarithmic Functions Example B. Rewrite the log-form into the exp-form. a. log3(1/9) = –2 3-2 = 1/9 b. 2w = logv(a – b) v2w = a – b Example A. Rewrite the exp-form into the log-form. a. 42 = 16 log4(16) = 2 b. w = u2+v logu(w) = 2+v To convert the exp-form to the log–form: b = y x logb( y ) = x→ To convert the log–form to the exp–form: b = y x logb( y ) = x→
- 52. The Logarithmic Functions Example B. Rewrite the log-form into the exp-form. a. log3(1/9) = –2 3-2 = 1/9 b. 2w = logv(a – b) v2w = a – b Example A. Rewrite the exp-form into the log-form. a. 42 = 16 log4(16) = 2 b. w = u2+v logu(w) = 2+v To convert the exp-form to the log–form: b = y x logb( y ) = x→ To convert the log–form to the exp–form: b = y x logb( y ) = x→
- 53. The Logarithmic Functions Example B. Rewrite the log-form into the exp-form. a. log3(1/9) = –2 3-2 = 1/9 b. 2w = logv(a – b) v2w = a – b Example A. Rewrite the exp-form into the log-form. a. 42 = 16 log4(16) = 2 b. w = u2+v logu(w) = 2+v To convert the exp-form to the log–form: b = y x logb( y ) = x→ To convert the log–form to the exp–form: b = y x logb( y ) = x→
- 54. The Logarithmic Functions Example B. Rewrite the log-form into the exp-form. a. log3(1/9) = –2 3-2 = 1/9 b. 2w = logv(a – b) v2w = a – b Example A. Rewrite the exp-form into the log-form. a. 42 = 16 log4(16) = 2 b. w = u2+v logu(w) = 2+v To convert the exp-form to the log–form: b = y x logb( y ) = x→ To convert the log–form to the exp–form: b = y x logb( y ) = x→ The output of logb(x), i.e. the exponent in the defined relation, may be positive or negative.
- 55. The Logarithmic Functions Example C. a. Rewrite the exp-form into the log-form. 4–3 = 1/64 8–2 = 1/64 log4(1/64) = –3 log8(1/64) = –2 exp–form log–form b. Rewrite the log-form into the exp-form. (1/2)–2 = 4log1/2(4) = –2 log1/2(8) = –3 exp–formlog–form (1/2)–3 = 8
- 56. The Logarithmic Functions The Common Log and the Natural Log Example C. a. Rewrite the exp-form into the log-form. 4–3 = 1/64 8–2 = 1/64 log4(1/64) = –3 log8(1/64) = –2 exp–form log–form b. Rewrite the log-form into the exp-form. (1/2)–2 = 4log1/2(4) = –2 log1/2(8) = –3 exp–formlog–form (1/2)–3 = 8
- 57. The Logarithmic Functions Base 10 is called the common base. The Common Log and the Natural Log Example C. a. Rewrite the exp-form into the log-form. 4–3 = 1/64 8–2 = 1/64 log4(1/64) = –3 log8(1/64) = –2 exp–form log–form b. Rewrite the log-form into the exp-form. (1/2)–2 = 4log1/2(4) = –2 log1/2(8) = –3 exp–formlog–form (1/2)–3 = 8
- 58. The Logarithmic Functions Base 10 is called the common base. Log with base10,. written as log(x) without the base number b, is called the common log, The Common Log and the Natural Log Example C. a. Rewrite the exp-form into the log-form. 4–3 = 1/64 8–2 = 1/64 log4(1/64) = –3 log8(1/64) = –2 exp–form log–form b. Rewrite the log-form into the exp-form. (1/2)–2 = 4log1/2(4) = –2 log1/2(8) = –3 exp–formlog–form (1/2)–3 = 8
- 59. The Logarithmic Functions Base 10 is called the common base. Log with base10,. written as log(x) without the base number b, is called the common log, i.e. log(x) is log10(x). The Common Log and the Natural Log Example C. a. Rewrite the exp-form into the log-form. 4–3 = 1/64 8–2 = 1/64 log4(1/64) = –3 log8(1/64) = –2 exp–form log–form b. Rewrite the log-form into the exp-form. (1/2)–2 = 4log1/2(4) = –2 log1/2(8) = –3 exp–formlog–form (1/2)–3 = 8
- 60. Base e is called the natural base. The Common Log and the Natural Log
- 61. Base e is called the natural base. Log with base e is written as ln(x) and it’s called the natural log, The Common Log and the Natural Log
- 62. Base e is called the natural base. Log with base e is written as ln(x) and it’s called the natural log, i.e. In(x) is loge(x). The Common Log and the Natural Log
- 63. Base e is called the natural base. Log with base e is written as ln(x) and it’s called the natural log, i.e. In(x) is loge(x). The Common Log and the Natural Log Example D. Convert to the other form. exp-form log-form 103 = 1000 ln(1/e2) = -2 ert = log(1) = 0 A P
- 64. Base e is called the natural base. Log with base e is written as ln(x) and it’s called the natural log, i.e. In(x) is loge(x). The Common Log and the Natural Log Example D. Convert to the other form. exp-form log-form 103 = 1000 log(1000) = 3 ln(1/e2) = -2 ert = log(1) = 0 A P
- 65. Base e is called the natural base. Log with base e is written as ln(x) and it’s called the natural log, i.e. In(x) is loge(x). The Common Log and the Natural Log Example D. Convert to the other form. exp-form log-form 103 = 1000 log(1000) = 3 e-2 = 1/e2 ln(1/e2) = -2 ert = log(1) = 0 A P
- 66. Base e is called the natural base. Log with base e is written as ln(x) and it’s called the natural log, i.e. In(x) is loge(x). The Common Log and the Natural Log Example D. Convert to the other form. exp-form log-form 103 = 1000 log(1000) = 3 e-2 = 1/e2 ln(1/e2) = -2 ert = ln( ) = rt log(1) = 0 A P A P
- 67. Base e is called the natural base. Log with base e is written as ln(x) and it’s called the natural log, i.e. In(x) is loge(x). The Common Log and the Natural Log Example D. Convert to the other form. exp-form log-form 103 = 1000 log(1000) = 3 e-2 = 1/e2 ln(1/e2) = -2 ert = ln( ) = rt 100 = 1 log(1) = 0 A P A P
- 68. Base e is called the natural base. Log with base e is written as ln(x) and it’s called the natural log, i.e. In(x) is loge(x). The Common Log and the Natural Log Example D. Convert to the other form. exp-form log-form 103 = 1000 log(1000) = 3 e-2 = 1/e2 ln(1/e2) = -2 ert = ln( ) = rt 100 = 1 log(1) = 0 A P A P Most log and powers can’t be computed efficiently by hand.
- 69. Base e is called the natural base. Log with base e is written as ln(x) and it’s called the natural log, i.e. In(x) is loge(x). The Common Log and the Natural Log Example D. Convert to the other form. exp-form log-form 103 = 1000 log(1000) = 3 e-2 = 1/e2 ln(1/e2) = -2 ert = ln( ) = rt 100 = 1 log(1) = 0 A P A P Most log and powers can’t be computed efficiently by hand. We need a calculation device to extract numerical solutions.
- 70. The Common Log and the Natural Log Example E. Calculate each of the following logs using a calculator. Then convert the relation into the exp–form and confirm the exp–form with a calculator. a. log(50) =
- 71. The Common Log and the Natural Log Example E. Calculate each of the following logs using a calculator. Then convert the relation into the exp–form and confirm the exp–form with a calculator. a. log(50) = 1.69897...
- 72. The Common Log and the Natural Log Example E. Calculate each of the following logs using a calculator. Then convert the relation into the exp–form and confirm the exp–form with a calculator. a. log(50) = 1.69897... In the exp–form, it’s101.69897 =
- 73. The Common Log and the Natural Log Example E. Calculate each of the following logs using a calculator. Then convert the relation into the exp–form and confirm the exp–form with a calculator. a. log(50) = 1.69897... In the exp–form, it’s101.69897 = 49.9999995...≈50
- 74. The Common Log and the Natural Log Example E. Calculate each of the following logs using a calculator. Then convert the relation into the exp–form and confirm the exp–form with a calculator. a. log(50) = 1.69897... In the exp–form, it’s101.69897 = 49.9999995...≈50 b. ln(9) =
- 75. The Common Log and the Natural Log Example E. Calculate each of the following logs using a calculator. Then convert the relation into the exp–form and confirm the exp–form with a calculator. a. log(50) = 1.69897... In the exp–form, it’s101.69897 = 49.9999995...≈50 b. ln(9) = 2.1972245..
- 76. The Common Log and the Natural Log Example E. Calculate each of the following logs using a calculator. Then convert the relation into the exp–form and confirm the exp–form with a calculator. a. log(50) = 1.69897... In the exp–form, it’s101.69897 = 49.9999995...≈50 b. ln(9) = 2.1972245.. In the exp–form, it’s e2.1972245 =
- 77. The Common Log and the Natural Log Example E. Calculate each of the following logs using a calculator. Then convert the relation into the exp–form and confirm the exp–form with a calculator. a. log(50) = 1.69897... In the exp–form, it’s101.69897 = 49.9999995...≈50 b. ln(9) = 2.1972245.. In the exp–form, it’s e2.1972245 = 8.9999993...≈ 9
- 78. The Common Log and the Natural Log Example E. Calculate each of the following logs using a calculator. Then convert the relation into the exp–form and confirm the exp–form with a calculator. a. log(50) = 1.69897... In the exp–form, it’s101.69897 = 49.9999995...≈50 b. ln(9) = 2.1972245.. c. Calculate the power using a calculator. Then convert the relation into the log–form and confirm the log–form by the calculator. e4.3 = In the exp–form, it’s e2.1972245 = 8.9999993...≈ 9
- 79. The Common Log and the Natural Log Example E. Calculate each of the following logs using a calculator. Then convert the relation into the exp–form and confirm the exp–form with a calculator. a. log(50) = 1.69897... In the exp–form, it’s101.69897 = 49.9999995...≈50 b. ln(9) = 2.1972245.. c. Calculate the power using a calculator. Then convert the relation into the log–form and confirm the log–form by the calculator. e4.3 = 73.699793.. In the exp–form, it’s e2.1972245 = 8.9999993...≈ 9
- 80. The Common Log and the Natural Log Example E. Calculate each of the following logs using a calculator. Then convert the relation into the exp–form and confirm the exp–form with a calculator. a. log(50) = 1.69897... In the exp–form, it’s101.69897 = 49.9999995...≈50 b. ln(9) = 2.1972245.. c. Calculate the power using a calculator. Then convert the relation into the log–form and confirm the log–form by the calculator. e4.3 = 73.699793..→ In(73.699793) = In the exp–form, it’s e2.1972245 = 8.9999993...≈ 9
- 81. The Common Log and the Natural Log Example E. Calculate each of the following logs using a calculator. Then convert the relation into the exp–form and confirm the exp–form with a calculator. a. log(50) = 1.69897... In the exp–form, it’s101.69897 = 49.9999995...≈50 b. ln(9) = 2.1972245.. c. Calculate the power using a calculator. Then convert the relation into the log–form and confirm the log–form by the calculator. e4.3 = 73.699793..→ In(73.699793) = 4.299999..≈ 4.3 In the exp–form, it’s e2.1972245 = 8.9999993...≈ 9
- 82. The Common Log and the Natural Log Example E. Calculate each of the following logs using a calculator. Then convert the relation into the exp–form and confirm the exp–form with a calculator. a. log(50) = 1.69897... In the exp–form, it’s101.69897 = 49.9999995...≈50 b. ln(9) = 2.1972245.. c. Calculate the power using a calculator. Then convert the relation into the log–form and confirm the log–form by the calculator. e4.3 = 73.699793..→ In(73.699793) = 4.299999..≈ 4.3 Your turn. Follow the instructions in part c for 10π. In the exp–form, it’s e2.1972245 = 8.9999993...≈ 9
- 83. Equation may be formed with log–notation. The Common Log and the Natural Log
- 84. Equation may be formed with log–notation. Often we need to restate them in the exp–form. The Common Log and the Natural Log
- 85. Equation may be formed with log–notation. Often we need to restate them in the exp–form. We say we "drop the log" when this step is taken. The Common Log and the Natural Log
- 86. Example F. Solve for x a. log9(x) = –1 Equation may be formed with log–notation. Often we need to restate them in the exp–form. We say we "drop the log" when this step is taken. The Common Log and the Natural Log
- 87. Example F. Solve for x a. log9(x) = –1 Drop the log and get x = 9–1. Equation may be formed with log–notation. Often we need to restate them in the exp–form. We say we "drop the log" when this step is taken. The Common Log and the Natural Log
- 88. Example F. Solve for x a. log9(x) = –1 Drop the log and get x = 9–1. So x = 1/9 Equation may be formed with log–notation. Often we need to restate them in the exp–form. We say we "drop the log" when this step is taken. The Common Log and the Natural Log
- 89. Example F. Solve for x a. log9(x) = –1 Drop the log and get x = 9–1. So x = 1/9 b. logx(9) = –2 Equation may be formed with log–notation. Often we need to restate them in the exp–form. We say we "drop the log" when this step is taken. The Common Log and the Natural Log
- 90. Example F. Solve for x a. log9(x) = –1 Drop the log and get x = 9–1. So x = 1/9 b. logx(9) = –2 Drop the log and get 9 = x–2, Equation may be formed with log–notation. Often we need to restate them in the exp–form. We say we "drop the log" when this step is taken. The Common Log and the Natural Log
- 91. Example F. Solve for x a. log9(x) = –1 Drop the log and get x = 9–1. So x = 1/9 b. logx(9) = –2 Drop the log and get 9 = x–2, i.e. 9 = 1 x2 Equation may be formed with log–notation. Often we need to restate them in the exp–form. We say we "drop the log" when this step is taken. The Common Log and the Natural Log
- 92. Example F. Solve for x a. log9(x) = –1 Drop the log and get x = 9–1. So x = 1/9 b. logx(9) = –2 Drop the log and get 9 = x–2, i.e. 9 = So 9x2 = 1 1 x2 Equation may be formed with log–notation. Often we need to restate them in the exp–form. We say we "drop the log" when this step is taken. The Common Log and the Natural Log
- 93. Example F. Solve for x a. log9(x) = –1 Drop the log and get x = 9–1. So x = 1/9 b. logx(9) = –2 Drop the log and get 9 = x–2, i.e. 9 = So 9x2 = 1 x2 = 1/9 x = 1/3 or x= –1/3 1 x2 Equation may be formed with log–notation. Often we need to restate them in the exp–form. We say we "drop the log" when this step is taken. The Common Log and the Natural Log
- 94. Example F. Solve for x a. log9(x) = –1 Drop the log and get x = 9–1. So x = 1/9 b. logx(9) = –2 Drop the log and get 9 = x–2, i.e. 9 = So 9x2 = 1 x2 = 1/9 x = 1/3 or x= –1/3 Since the base b > 0, so x = 1/3 is the only solution. 1 x2 Equation may be formed with log–notation. Often we need to restate them in the exp–form. We say we "drop the log" when this step is taken. The Common Log and the Natural Log
- 95. The Logarithmic Functions Graphs of the Logarithmic Functions Recall that the domain of logb(x) is the set of all x > 0.
- 96. The Logarithmic Functions Graphs of the Logarithmic Functions 1/4 1/2 1 2 4 8 x y=log2(x) Recall that the domain of logb(x) is the set of all x > 0. Hence to make a table to plot the graph of y = log2(x), we only select positive x’s.
- 97. The Logarithmic Functions Graphs of the Logarithmic Functions 2 4 8 x y=log2(x) Recall that the domain of logb(x) is the set of all x > 0. Hence to make a table to plot the graph of y = log2(x), we only select positive x’s. In particular we select x’s related to base 2 for easy computation of the y’s.
- 98. The Logarithmic Functions Graphs of the Logarithmic Functions 1/4 1/2 1 2 4 8 x y=log2(x) Recall that the domain of logb(x) is the set of all x > 0. Hence to make a table to plot the graph of y = log2(x), we only select positive x’s. In particular we select x’s related to base 2 for easy computation of the y’s.
- 99. The Logarithmic Functions Graphs of the Logarithmic Functions 1/4 -2 1/2 1 2 4 8 x y=log2(x) Recall that the domain of logb(x) is the set of all x > 0. Hence to make a table to plot the graph of y = log2(x), we only select positive x’s. In particular we select x’s related to base 2 for easy computation of the y’s.
- 100. The Logarithmic Functions Graphs of the Logarithmic Functions 1/4 -2 1/2 -1 1 2 4 8 x y=log2(x) Recall that the domain of logb(x) is the set of all x > 0. Hence to make a table to plot the graph of y = log2(x), we only select positive x’s. In particular we select x’s related to base 2 for easy computation of the y’s.
- 101. The Logarithmic Functions Graphs of the Logarithmic Functions 1/4 -2 1/2 -1 1 0 2 4 8 x y=log2(x) Recall that the domain of logb(x) is the set of all x > 0. Hence to make a table to plot the graph of y = log2(x), we only select positive x’s. In particular we select x’s related to base 2 for easy computation of the y’s.
- 102. The Logarithmic Functions Graphs of the Logarithmic Functions 1/4 -2 1/2 -1 1 0 2 1 4 8 x y=log2(x) Recall that the domain of logb(x) is the set of all x > 0. Hence to make a table to plot the graph of y = log2(x), we only select positive x’s. In particular we select x’s related to base 2 for easy computation of the y’s.
- 103. The Logarithmic Functions Graphs of the Logarithmic Functions 1/4 -2 1/2 -1 1 0 2 1 4 2 8 3 x y=log2(x) Recall that the domain of logb(x) is the set of all x > 0. Hence to make a table to plot the graph of y = log2(x), we only select positive x’s. In particular we select x’s related to base 2 for easy computation of the y’s.
- 104. The Logarithmic Functions (1, 0) (2, 1) (4, 2) (8, 3) (16, 4) (1/2, -1) (1/4, -2) y=log2(x) Graphs of the Logarithmic Functions 1/4 -2 1/2 -1 1 0 2 1 4 2 8 3 x y=log2(x) Recall that the domain of logb(x) is the set of all x > 0. Hence to make a table to plot the graph of y = log2(x), we only select positive x’s. In particular we select x’s related to base 2 for easy computation of the y’s. x y
- 105. The Logarithmic Functions To graph log with base b = ½, we have log1/2(4) = –2, log1/2(8) = –3, log1/2(16) = –4
- 106. The Logarithmic Functions x y (1, 0) (8, -3) To graph log with base b = ½, we have log1/2(4) = –2, log1/2(8) = –3, log1/2(16) = –4 (4, -2) (16, -4) y = log1/2(x)
- 107. The Logarithmic Functions x y (1, 0) (8, -3) To graph log with base b = ½, we have log1/2(4) = –2, log1/2(8) = –3, log1/2(16) = –4 (4, -2) (16, -4) y = log1/2(x) x x y (1, 0)(1, 0) y = logb(x), b > 1 y = logb(x), 1 > b Here are the general shapes of log–functions. y (b, 1) (b, 1)
- 108. 1. b0 = 1 2. br · bt = br+t 3. = br-t 4. (br)t = brt bt br Properties of Logarithm Recall the following Rules of Exponents: The corresponding Rules of Logs are:
- 109. 1. logb(1) = 01. b0 = 1 2. br · bt = br+t 3. = br-t 4. (br)t = brt bt br Properties of Logarithm Recall the following Rules of Exponents: The corresponding Rules of Logs are:
- 110. 1. logb(1) = 0 2. logb(x·y) = logb(x)+logb(y) 1. b0 = 1 2. br · bt = br+t 3. = br-t 4. (br)t = brt bt br Properties of Logarithm Recall the following Rules of Exponents: The corresponding Rules of Logs are:
- 111. 1. logb(1) = 0 2. logb(x·y) = logb(x)+logb(y) 3. logb( ) = logb(x) – logb(y)x y 1. b0 = 1 2. br · bt = br+t 3. = br-t 4. (br)t = brt bt br Properties of Logarithm Recall the following Rules of Exponents: The corresponding Rules of Logs are:
- 112. 1. logb(1) = 0 2. logb(x·y) = logb(x)+logb(y) 3. logb( ) = logb(x) – logb(y) 4. logb(xt) = t·logb(x) x y 1. b0 = 1 2. br · bt = br+t 3. = br-t 4. (br)t = brt bt br Properties of Logarithm Recall the following Rules of Exponents: The corresponding Rules of Logs are:
- 113. 1. logb(1) = 0 2. logb(x·y) = logb(x)+logb(y) 3. logb( ) = logb(x) – logb(y) 4. logb(xt) = t·logb(x) x y 1. b0 = 1 2. br · bt = br+t 3. = br-t 4. (br)t = brt bt br Properties of Logarithm Recall the following Rules of Exponents: The corresponding Rules of Logs are: We veryify part 2: logb(xy) = logb(x) + logb(y), x, y > 0. Proof:
- 114. 1. logb(1) = 0 2. logb(x·y) = logb(x)+logb(y) 3. logb( ) = logb(x) – logb(y) 4. logb(xt) = t·logb(x) x y 1. b0 = 1 2. br · bt = br+t 3. = br-t 4. (br)t = brt bt br Properties of Logarithm Recall the following Rules of Exponents: The corresponding Rules of Logs are: We veryify part 2: logb(xy) = logb(x) + logb(y), x, y > 0. Proof: Let x and y be two positive numbers.
- 115. 1. logb(1) = 0 2. logb(x·y) = logb(x)+logb(y) 3. logb( ) = logb(x) – logb(y) 4. logb(xt) = t·logb(x) x y 1. b0 = 1 2. br · bt = br+t 3. = br-t 4. (br)t = brt bt br Properties of Logarithm Recall the following Rules of Exponents: The corresponding Rules of Logs are: We veryify part 2: logb(xy) = logb(x) + logb(y), x, y > 0. Proof: Let x and y be two positive numbers. Let logb(x) = r and logb(y) = t, which in exp-form are x = br and y = bt.
- 116. 1. logb(1) = 0 2. logb(x·y) = logb(x)+logb(y) 3. logb( ) = logb(x) – logb(y) 4. logb(xt) = t·logb(x) x y 1. b0 = 1 2. br · bt = br+t 3. = br-t 4. (br)t = brt bt br Properties of Logarithm Recall the following Rules of Exponents: The corresponding Rules of Logs are: We veryify part 2: logb(xy) = logb(x) + logb(y), x, y > 0. Proof: Let x and y be two positive numbers. Let logb(x) = r and logb(y) = t, which in exp-form are x = br and y = bt. Therefore x·y = br+t,
- 117. 1. logb(1) = 0 2. logb(x·y) = logb(x)+logb(y) 3. logb( ) = logb(x) – logb(y) 4. logb(xt) = t·logb(x) x y 1. b0 = 1 2. br · bt = br+t 3. = br-t 4. (br)t = brt bt br Properties of Logarithm Recall the following Rules of Exponents: The corresponding Rules of Logs are: We veryify part 2: logb(xy) = logb(x) + logb(y), x, y > 0. Proof: Let x and y be two positive numbers. Let logb(x) = r and logb(y) = t, which in exp-form are x = br and y = bt. Therefore x·y = br+t, which in log-form is logb(x·y) = r + t = logb(x)+logb(y).
- 118. 1. logb(1) = 0 2. logb(x·y) = logb(x)+logb(y) 3. logb( ) = logb(x) – logb(y) 4. logb(xt) = t·logb(x) x y 1. b0 = 1 2. br · bt = br+t 3. = br-t 4. (br)t = brt bt br Properties of Logarithm Recall the following Rules of Exponents: The corresponding Rules of Logs are: We veryify part 2: logb(xy) = logb(x) + logb(y), x, y > 0. Proof: Let x and y be two positive numbers. Let logb(x) = r and logb(y) = t, which in exp-form are x = br and y = bt. Therefore x·y = br+t, which in log-form is logb(x·y) = r + t = logb(x)+logb(y). The other rules may be verified similarly.
- 119. Example G. 3x2 y Properties of Logarithm a. Write log( ) in terms of log(x) and log(y).
- 120. 3x2 y log( ) = log( ),3x2 y 3x2 y1/2 Properties of Logarithm a. Write log( ) in terms of log(x) and log(y). Example G.
- 121. 3x2 y log( ) = log( ), by the quotient rule = log (3x2) – log(y1/2) 3x2 y 3x2 y1/2 Properties of Logarithm a. Write log( ) in terms of log(x) and log(y). Example G.
- 122. 3x2 y log( ) = log( ), by the quotient rule = log (3x2) – log(y1/2) product rule = log(3) + log(x2) 3x2 y 3x2 y1/2 Properties of Logarithm a. Write log( ) in terms of log(x) and log(y). Example G.
- 123. 3x2 y log( ) = log( ), by the quotient rule = log (3x2) – log(y1/2) product rule power rule = log(3) + log(x2) – ½ log(y) 3x2 y 3x2 y1/2 Properties of Logarithm a. Write log( ) in terms of log(x) and log(y). Example G.
- 124. 3x2 y log( ) = log( ), by the quotient rule = log (3x2) – log(y1/2) product rule power rule = log(3) + log(x2) – ½ log(y) = log(3) + 2log(x) – ½ log(y) 3x2 y 3x2 y1/2 Properties of Logarithm a. Write log( ) in terms of log(x) and log(y). Example G.
- 125. 3x2 y log( ) = log( ), by the quotient rule = log (3x2) – log(y1/2) product rule power rule = log(3) + log(x2) – ½ log(y) = log(3) + 2log(x) – ½ log(y) 3x2 y 3x2 y1/2 Properties of Logarithm a. Write log( ) in terms of log(x) and log(y). b. Combine log(3) + 2log(x) – ½ log(y) into one log. Example G.
- 126. 3x2 y log( ) = log( ), by the quotient rule = log (3x2) – log(y1/2) product rule power rule = log(3) + log(x2) – ½ log(y) = log(3) + 2log(x) – ½ log(y) 3x2 y 3x2 y1/2 Properties of Logarithm a. Write log( ) in terms of log(x) and log(y). log(3) + 2log(x) – ½ log(y) power rule = log(3) + log(x2) – log(y1/2) b. Combine log(3) + 2log(x) – ½ log(y) into one log. Example G.
- 127. 3x2 y log( ) = log( ), by the quotient rule = log (3x2) – log(y1/2) product rule power rule = log(3) + log(x2) – ½ log(y) = log(3) + 2log(x) – ½ log(y) 3x2 y 3x2 y1/2 Properties of Logarithm a. Write log( ) in terms of log(x) and log(y). log(3) + 2log(x) – ½ log(y) power rule = log(3) + log(x2) – log(y1/2) product rule = log (3x2) – log(y1/2) b. Combine log(3) + 2log(x) – ½ log(y) into one log. Example G.
- 128. 3x2 y log( ) = log( ), by the quotient rule = log (3x2) – log(y1/2) product rule power rule = log(3) + log(x2) – ½ log(y) = log(3) + 2log(x) – ½ log(y) 3x2 y 3x2 y1/2 Properties of Logarithm a. Write log( ) in terms of log(x) and log(y). log(3) + 2log(x) – ½ log(y) power rule = log(3) + log(x2) – log(y1/2) product rule = log (3x2) – log(y1/2)= log( )3x2 y1/2 b. Combine log(3) + 2log(x) – ½ log(y) into one log. Example G.
- 129. Recall that given a pair of inverse functions, f and f -1, then f(f -1(x)) = x and f -1(f(x)) = x. Properties of Logarithm
- 130. Recall that given a pair of inverse functions, f and f -1, then f(f -1(x)) = x and f -1(f(x)) = x. Since expb(x) and logb(x) is a pair of inverse functions, Properties of Logarithm
- 131. Recall that given a pair of inverse functions, f and f -1, then f(f -1(x)) = x and f -1(f(x)) = x. Since expb(x) and logb(x) is a pair of inverse functions, we have that: a. logb(expb(x)) = x or logb(bx) = x Properties of Logarithm
- 132. Recall that given a pair of inverse functions, f and f -1, then f(f -1(x)) = x and f -1(f(x)) = x. Since expb(x) and logb(x) is a pair of inverse functions, we have that: a. logb(expb(x)) = x or logb(bx) = x b. expb(logb(x)) = x or blog (x) = x Properties of Logarithm b
- 133. Recall that given a pair of inverse functions, f and f -1, then f(f -1(x)) = x and f -1(f(x)) = x. Since expb(x) and logb(x) is a pair of inverse functions, we have that: a. logb(expb(x)) = x or logb(bx) = x b. expb(logb(x)) = x or blog (x) = x Properties of Logarithm b Example H. Simplify a. log2(2-5) = b. 8log (xy) = c. e2+ln(7) = 8
- 134. Recall that given a pair of inverse functions, f and f -1, then f(f -1(x)) = x and f -1(f(x)) = x. Since expb(x) and logb(x) is a pair of inverse functions, we have that: a. logb(expb(x)) = x or logb(bx) = x b. expb(logb(x)) = x or blog (x) = x Properties of Logarithm b Example H. Simplify a. log2(2-5) = -5 b. 8log (xy) = c. e2+ln(7) = 8
- 135. Recall that given a pair of inverse functions, f and f -1, then f(f -1(x)) = x and f -1(f(x)) = x. Since expb(x) and logb(x) is a pair of inverse functions, we have that: a. logb(expb(x)) = x or logb(bx) = x b. expb(logb(x)) = x or blog (x) = x Properties of Logarithm b Example H. Simplify a. log2(2-5) = -5 b. 8log (xy) = xy c. e2+ln(7) = 8
- 136. Recall that given a pair of inverse functions, f and f -1, then f(f -1(x)) = x and f -1(f(x)) = x. Since expb(x) and logb(x) is a pair of inverse functions, we have that: a. logb(expb(x)) = x or logb(bx) = x b. expb(logb(x)) = x or blog (x) = x Properties of Logarithm b Example H. Simplify a. log2(2-5) = -5 b. 8log (xy) = xy c. e2+ln(7) = e2·eln(7) = 7e2 8