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- 1. The Logarithmic Functions
- 2. The Logarithmic Functions yFrom the graph of y = 10 we see x y = 10xthat the output y encompasses allthe positive numbers. x
- 3. The Logarithmic Functions yFrom the graph of y = 10 we see x y = 10xthat the output y encompasses allthe positive numbers. x
- 4. The Logarithmic Functions yFrom the graph of y = 10 we see x y = 10xthat the output y encompasses allthe positive numbers. xIn fact, given any positive number y,there is exactly one x such that 10x = y.
- 5. The Logarithmic Functions yFrom the graph of y = 10 we see x y = 10xthat the output y encompasses allthe positive numbers. xIn fact, given any positive number y,there is exactly one x such that 10x = y.For example, if y is 100,
- 6. The Logarithmic Functions yFrom the graph of y = 10 we see x y = 10xthat the output y encompasses allthe positive numbers. xIn fact, given any positive number y,there is exactly one x such that 10x = y.For example, if y is 100, since100 = 102then x must be 2,
- 7. The Logarithmic Functions yFrom the graph of y = 10 we see x y = 10 xthat the output y encompasses allthe positive numbers. xIn fact, given any positive number y,there is exactly one x such that 10x = y.For example, if y is 100, since100 = 102then x must be 2, i.e. log(100) = 2.Similarly if y is 5, then x must be log(5) = 0.6989..since 5 = 100.6989...
- 8. The Logarithmic Functions yFrom the graph of y = 10 we see x y = 10 xthat the output y encompasses allthe positive numbers. xIn fact, given any positive number y,there is exactly one x such that 10x = y.For example, if y is 100, since100 = 102then x must be 2, i.e. log(100) = 2.Similarly if y is 5, then x must be log(5) = 0.6989..since 5 = 100.6989... So log(y) is a well defined function.
- 9. The Logarithmic Functions yFrom the graph of y = 10 we see x y = 10 xthat the output y encompasses allthe positive numbers. xIn fact, given any positive number y,there is exactly one x such that 10x = y.For example, if y is 100, since100 = 102then x must be 2, i.e. log(100) = 2.Similarly if y is 5, then x must be log(5) = 0.6989..since 5 = 100.6989... So log(y) is a well defined function.This is also the case for all other positive bases b .
- 10. The Logarithmic Functions yFrom the graph of y = 10 we see x y = 10 xthat the output y encompasses allthe positive numbers. xIn fact, given any positive number y,there is exactly one x such that 10x = y.For example, if y is 100, since100 = 102then x must be 2, i.e. log(100) = 2.Similarly if y is 5, then x must be log(5) = 0.6989..since 5 = 100.6989... So log(y) is a well defined function.This is also the case for all other positive bases b .The Existence of logb( y )
- 11. The Logarithmic Functions yFrom the graph of y = 10 we see x y = 10 xthat the output y encompasses allthe positive numbers. xIn fact, given any positive number y,there is exactly one x such that 10x = y.For example, if y is 100, since100 = 102then x must be 2, i.e. log(100) = 2.Similarly if y is 5, then x must be log(5) = 0.6989..since 5 = 100.6989... So log(y) is a well defined function.This is also the case for all other positive bases b .The Existence of logb( y )Given any positive base b (b ≠ 1) and any positivenumber y, there is exactly one x such that y = bxi.e. x = logb(y) is well defined.
- 12. Properties of Logarithm
- 13. Properties of LogarithmRecall the following The correspondingRules of Exponents: Rules of Logs are:
- 14. Properties of LogarithmRecall the following The correspondingRules of Exponents: Rules of Logs are:1. b0 = 1
- 15. Properties of LogarithmRecall the following The correspondingRules of Exponents: Rules of Logs are:1. b0 = 1 1. logb(1) = 0
- 16. Properties of LogarithmRecall the following The correspondingRules of Exponents: Rules of Logs are:1. b0 = 1 1. logb(1) = 02. br · bt = br+t
- 17. Properties of LogarithmRecall the following The correspondingRules of Exponents: Rules of Logs are:1. b0 = 1 1. logb(1) = 02. br · bt = br+t 2. logb(x·y) = logb(x)+logb(y)
- 18. Properties of LogarithmRecall the following The correspondingRules of Exponents: Rules of Logs are:1. b0 = 1 1. logb(1) = 02. br · bt = br+t 2. logb(x·y) = logb(x)+logb(y) In this version, logb(x) corresponds to r, logb(y) corresponds to t.
- 19. Properties of LogarithmRecall the following The correspondingRules of Exponents: Rules of Logs are:1. b0 = 1 1. logb(1) = 02. br · bt = br+t 2. logb(x·y) = logb(x)+logb(y) br = br-t3. t b
- 20. Properties of LogarithmRecall the following The correspondingRules of Exponents: Rules of Logs are:1. b0 = 1 1. logb(1) = 02. br · bt = br+t 2. logb(x·y) = logb(x)+logb(y) br = br-t3. t x b 3. logb( y ) = logb(x) – logb(y)
- 21. Properties of LogarithmRecall the following The correspondingRules of Exponents: Rules of Logs are:1. b0 = 1 1. logb(1) = 02. br · bt = br+t 2. logb(x·y) = logb(x)+logb(y)3. tbr = br-t x b 3. logb( y ) = logb(x) – logb(y)4. (br)t = brt
- 22. Properties of LogarithmRecall the following The correspondingRules of Exponents: Rules of Logs are:1. b0 = 1 1. logb(1) = 02. br · bt = br+t 2. logb(x·y) = logb(x)+logb(y)3. tbr = br-t x b 3. logb( y ) = logb(x) – logb(y)4. (br)t = brt 4. logb(xt) = t·logb(x)
- 23. Properties of LogarithmRecall the following The correspondingRules of Exponents: Rules of Logs are:1. b0 = 1 1. logb(1) = 02. br · bt = br+t 2. logb(x·y) = logb(x)+logb(y)3. tbr = br-t x b 3. logb( y ) = logb(x) – logb(y)4. (br)t = brt 4. logb(xt) = t·logb(x)We veryify part 2: logb(xy) = logb(x) + logb(y), x, y > 0.Proof:
- 24. Properties of LogarithmRecall the following The correspondingRules of Exponents: Rules of Logs are:1. b0 = 1 1. logb(1) = 02. br · bt = br+t 2. logb(x·y) = logb(x)+logb(y)3. tbr = br-t x b 3. logb( y ) = logb(x) – logb(y)4. (br)t = brt 4. logb(xt) = t·logb(x)We veryify part 2: logb(xy) = logb(x) + logb(y), x, y > 0.Proof:Let x and y be two positive numbers.
- 25. Properties of LogarithmRecall the following The correspondingRules of Exponents: Rules of Logs are:1. b0 = 1 1. logb(1) = 02. br · bt = br+t 2. logb(x·y) = logb(x)+logb(y)3. tbr = br-t x b 3. logb( y ) = logb(x) – logb(y)4. (br)t = brt 4. logb(xt) = t·logb(x)We veryify part 2: logb(xy) = logb(x) + logb(y), x, y > 0.Proof:Let x and y be two positive numbers. Let log b(x) = rand logb(y) = t, which in exp-form are x = br and y = bt.
- 26. Properties of LogarithmRecall the following The correspondingRules of Exponents: Rules of Logs are:1. b0 = 1 1. logb(1) = 02. br · bt = br+t 2. logb(x·y) = logb(x)+logb(y)3. tbr = br-t x b 3. logb( y ) = logb(x) – logb(y)4. (br)t = brt 4. logb(xt) = t·logb(x)We veryify part 2: logb(xy) = logb(x) + logb(y), x, y > 0.Proof:Let x and y be two positive numbers. Let log b(x) = rand logb(y) = t, which in exp-form are x = br and y = bt.Therefore x·y = br+t,
- 27. Properties of LogarithmRecall the following The correspondingRules of Exponents: Rules of Logs are:1. b0 = 1 1. logb(1) = 02. br · bt = br+t 2. logb(x·y) = logb(x)+logb(y)3. tbr = br-t x b 3. logb( y ) = logb(x) – logb(y)4. (br)t = brt 4. logb(xt) = t·logb(x)We veryify part 2: logb(xy) = logb(x) + logb(y), x, y > 0.Proof:Let x and y be two positive numbers. Let log b(x) = rand logb(y) = t, which in exp-form are x = br and y = bt.Therefore x·y = br+t, which in log-form islogb(x·y) = r + t = logb(x)+logb(y).
- 28. Properties of LogarithmRecall the following The correspondingRules of Exponents: Rules of Logs are:1. b0 = 1 1. logb(1) = 02. br · bt = br+t 2. logb(x·y) = logb(x)+logb(y)3. tbr = br-t x b 3. logb( y ) = logb(x) – logb(y)4. (br)t = brt 4. logb(xt) = t·logb(x)We veryify part 2: logb(xy) = logb(x) + logb(y), x, y > 0.Proof:Let x and y be two positive numbers. Let log b(x) = rand logb(y) = t, which in exp-form are x = br and y = bt.Therefore x·y = br+t, which in log-form isThe(x·y) = rules = logbbe verified similarly.logb other r + t may (x)+logb(y).
- 29. Properties of LogarithmExample A.a. Write log( 3x2 ) in terms of log(x) and log(y). √y
- 30. Properties of LogarithmExample A.a. Write log( 3x2 ) in terms of log(x) and log(y). √ylog( 3x2 ) = log( 3x2 ), √y y1/2
- 31. Properties of LogarithmExample A.a. Write log( 3x2 ) in terms of log(x) and log(y). √ylog( 3x2 ) = log( 3x2 ), by the quotient rule √y y1/2 = log (3x2) – log(y1/2)
- 32. Properties of LogarithmExample A.a. Write log( 3x2 ) in terms of log(x) and log(y). √ylog( 3x2 ) = log( 3x2 ), by the quotient rule √y y1/2 = log (3x2) – log(y1/2) product rule = log(3) + log(x2)
- 33. Properties of LogarithmExample A.a. Write log( 3x2 ) in terms of log(x) and log(y). √ylog( 3x2 ) = log( 3x2), by the quotient rule √y y1/2 = log (3x2) – log(y1/2) product rule power rule = log(3) + log(x2) – ½ log(y)
- 34. Properties of LogarithmExample A.a. Write log( 3x2 ) in terms of log(x) and log(y). √ylog( 3x2 ) = log( 3x2), by the quotient rule √y y1/2 = log (3x2) – log(y1/2) product rule power rule = log(3) + log(x2) – ½ log(y) = log(3) + 2log(x) – ½ log(y)
- 35. Properties of LogarithmExample A.a. Write log( 3x2 ) in terms of log(x) and log(y). √ylog( 3x2 ) = log( 3x2), by the quotient rule √y y1/2 = log (3x2) – log(y1/2) product rule power rule = log(3) + log(x2) – ½ log(y) = log(3) + 2log(x) – ½ log(y)b. Combine log(3) + 2log(x) – ½ log(y) into one log.
- 36. Properties of LogarithmExample A.a. Write log( 3x2 ) in terms of log(x) and log(y). √ylog( 3x2 ) = log( 3x2), by the quotient rule √y y1/2 = log (3x2) – log(y1/2) product rule power rule = log(3) + log(x2) – ½ log(y) = log(3) + 2log(x) – ½ log(y)b. Combine log(3) + 2log(x) – ½ log(y) into one log.log(3) + 2log(x) – ½ log(y) power rule= log(3) + log(x2) – log(y1/2)
- 37. Properties of LogarithmExample A.a. Write log( 3x2 ) in terms of log(x) and log(y). √ylog( 3x2 ) = log( 3x2), by the quotient rule √y y1/2 = log (3x2) – log(y1/2) product rule power rule = log(3) + log(x2) – ½ log(y) = log(3) + 2log(x) – ½ log(y)b. Combine log(3) + 2log(x) – ½ log(y) into one log.log(3) + 2log(x) – ½ log(y) power rule= log(3) + log(x2) – log(y1/2) product rule= log (3x2) – log(y1/2)
- 38. Properties of LogarithmExample A.a. Write log( 3x2 ) in terms of log(x) and log(y). √ylog( 3x2 ) = log( 3x2), by the quotient rule √y y1/2 = log (3x2) – log(y1/2) product rule power rule = log(3) + log(x2) – ½ log(y) = log(3) + 2log(x) – ½ log(y)b. Combine log(3) + 2log(x) – ½ log(y) into one log.log(3) + 2log(x) – ½ log(y) power rule= log(3) + log(x2) – log(y1/2) product rule 2 1/2 3x2)= log (3x ) – log(y )= log( 1/2 y
- 39. Properties of LogarithmThe exponential function bx is also written as expb(x).
- 40. Properties of LogarithmThe exponential function bx is also written as expb(x).For example, exp10(x) is 10x and exp10(2) = 102 = 100.
- 41. Properties of LogarithmThe exponential function bx is also written as expb(x).For example, exp10(x) is 10x and exp10(2) = 102 = 100.The pair of functions expb(x) and logb(x) scramble andunscramble the output of each other like a pair ofcoding–decoding machines.
- 42. Properties of LogarithmThe exponential function bx is also written as expb(x).For example, exp10(x) is 10x and exp10(2) = 102 = 100.The pair of functions expb(x) and logb(x) scramble andunscramble the output of each other like a pair ofcoding–decoding machines. An input x after beingprocessed by one function may be de–processed bythe other.
- 43. Properties of LogarithmThe exponential function bx is also written as expb(x).For example, exp10(x) is 10x and exp10(2) = 102 = 100.The pair of functions expb(x) and logb(x) scramble andunscramble the output of each other like a pair ofcoding–decoding machines. An input x after beingprocessed by one function may be de–processed bythe other. We will illustrate this with the pair exp10(x),log (x) and the relation 102 = 100.
- 44. Properties of LogarithmThe exponential function bx is also written as expb(x).For example, exp10(x) is 10x and exp10(2) = 102 = 100.The pair of functions expb(x) and logb(x) scramble andunscramble the output of each other like a pair ofcoding–decoding machines. An input x after beingprocessed by one function may be de–processed bythe other. We will illustrate this with the pair exp10(x),log (x) and the relation 102 = 100.Starting with an input, sayx=2
- 45. Properties of LogarithmThe exponential function bx is also written as expb(x).For example, exp10(x) is 10x and exp10(2) = 102 = 100.The pair of functions expb(x) and logb(x) scramble andunscramble the output of each other like a pair ofcoding–decoding machines. An input x after beingprocessed by one function may be de–processed bythe other. We will illustrate this with the pair exp10(x),log (x) and the relation 102 = 100.Starting with an input, sayx=2 exp (2) 10
- 46. Properties of LogarithmThe exponential function bx is also written as expb(x).For example, exp10(x) is 10x and exp10(2) = 102 = 100.The pair of functions expb(x) and logb(x) scramble andunscramble the output of each other like a pair ofcoding–decoding machines. An input x after beingprocessed by one function may be de–processed bythe other. We will illustrate this with the pair exp10(x),log (x) and the relation 102 = 100.Starting with an input, sayx=2 exp (2) 10 100
- 47. Properties of LogarithmThe exponential function bx is also written as expb(x).For example, exp10(x) is 10x and exp10(2) = 102 = 100.The pair of functions expb(x) and logb(x) scramble andunscramble the output of each other like a pair ofcoding–decoding machines. An input x after beingprocessed by one function may be de–processed bythe other. We will illustrate this with the pair exp10(x),log (x) and the relation 102 = 100.Starting with an input, sayx=2 exp (2) 10 100 log (100) 10
- 48. Properties of LogarithmThe exponential function bx is also written as expb(x).For example, exp10(x) is 10x and exp10(2) = 102 = 100.The pair of functions expb(x) and logb(x) scramble andunscramble the output of each other like a pair ofcoding–decoding machines. An input x after beingprocessed by one function may be de–processed bythe other. We will illustrate this with the pair exp10(x),log (x) and the relation 102 = 100.Starting with an input, sayx=2 exp (2) 10 100 log (100) 10 2 (the starting x)
- 49. Properties of LogarithmThe exponential function bx is also written as expb(x).For example, exp10(x) is 10x and exp10(2) = 102 = 100.The pair of functions expb(x) and logb(x) scramble andunscramble the output of each other like a pair ofcoding–decoding machines. An input x after beingprocessed by one function may be de–processed bythe other. We will illustrate this with the pair exp10(x),log (x) and the relation 102 = 100.Starting with an input, sayx=2 exp (2) 10 100 log (100) 10 2 (the starting x)x = 100
- 50. Properties of LogarithmThe exponential function bx is also written as expb(x).For example, exp10(x) is 10x and exp10(2) = 102 = 100.The pair of functions expb(x) and logb(x) scramble andunscramble the output of each other like a pair ofcoding–decoding machines. An input x after beingprocessed by one function may be de–processed bythe other. We will illustrate this with the pair exp10(x),log (x) and the relation 102 = 100.Starting with an input, sayx=2 exp (2) 10 100 log (100) 10 2 (the starting x)x = 100 log10(100)
- 51. Properties of LogarithmThe exponential function bx is also written as expb(x).For example, exp10(x) is 10x and exp10(2) = 102 = 100.The pair of functions expb(x) and logb(x) scramble andunscramble the output of each other like a pair ofcoding–decoding machines. An input x after beingprocessed by one function may be de–processed bythe other. We will illustrate this with the pair exp10(x),log (x) and the relation 102 = 100.Starting with an input, sayx=2 exp (2) 10 100 log (100) 10 2 (the starting x)x = 100 log10(100) 2
- 52. Properties of LogarithmThe exponential function bx is also written as expb(x).For example, exp10(x) is 10x and exp10(2) = 102 = 100.The pair of functions expb(x) and logb(x) scramble andunscramble the output of each other like a pair ofcoding–decoding machines. An input x after beingprocessed by one function may be de–processed bythe other. We will illustrate this with the pair exp10(x),log (x) and the relation 102 = 100.Starting with an input, sayx=2 exp (2) 10 100 log (100) 10 2 (the starting x)x = 100 log10(100) 2 exp10(2)
- 53. Properties of LogarithmThe exponential function bx is also written as expb(x).For example, exp10(x) is 10x and exp10(2) = 102 = 100.The pair of functions expb(x) and logb(x) scramble andunscramble the output of each other like a pair ofcoding–decoding machines. An input x after beingprocessed by one function may be de–processed bythe other. We will illustrate this with the pair exp10(x),log (x) and the relation 102 = 100.Starting with an input, sayx=2 exp (2) 10 100 log (100) 10 2 (the starting x)x = 100 log10(100) 2 exp10(2) 100 (the starting x)
- 54. Properties of LogarithmThe exponential function bx is also written as expb(x).For example, exp10(x) is 10x and exp10(2) = 102 = 100.The pair of functions expb(x) and logb(x) scramble andunscramble the output of each other like a pair ofcoding–decoding machines. An input x after beingprocessed by one function may be de–processed bythe other. We will illustrate this with the pair exp10(x),log (x) and the relation 102 = 100.Starting with an input, sayx=2 exp (2) 10 100 log (100) 10 2 (the starting x)x = 100 log10(100) 2 exp10(2) 100 (the starting x)A pair of functions such as expb(x) and logb(x) thatunscramble each other is called an inverse pair.
- 55. Properties of LogarithmSo for all pairs of expb(x) & logb(x) and an input x
- 56. Properties of LogarithmSo for all pairs of expb(x) & logb(x) and an input x x expb (x) #
- 57. Properties of LogarithmSo for all pairs of expb(x) & logb(x) and an input x x expb (x) # logb(#) x
- 58. Properties of LogarithmSo for all pairs of expb(x) & logb(x) and an input x x expb (x) # logb(#) xHere is the inverse relation stated in function notation.The Inverse Relation of Exp and Log b
- 59. Properties of LogarithmSo for all pairs of expb(x) & logb(x) and an input x x expb (x) # logb(#) xHere is the inverse relation stated in function notation.The Inverse Relation of Exp and Loga. logb(expb(x)) = x or logb(bx) = x b
- 60. Properties of LogarithmSo for all pairs of expb(x) & logb(x) and an input x x expb (x) # logb(#) x x (x > 0) logb (x) # expb(#) xHere is the inverse relation stated in function notation.The Inverse Relation of Exp and Loga. logb(expb(x)) = x or logb(bx) = x bb. expb(logb(x)) = x or blog (x) = x
- 61. Properties of LogarithmSo for all pairs of expb(x) & logb(x) and an input x x expb (x) # logb(#) x x (x > 0) logb (x) # expb(#) xHere is the inverse relation stated in function notation.The Inverse Relation of Exp and Loga. logb(expb(x)) = x or logb(bx) = x bb. expb(logb(x)) = x or blog (x) = xExample B: Simplifya. log2(2–5) =b. 8log 8(xy) =c. e2 + ln(7) =
- 62. Properties of LogarithmSo for all pairs of expb(x) & logb(x) and an input x x expb (x) # logb(#) x x (x > 0) logb (x) # expb(#) xHere is the inverse relation stated in function notation.The Inverse Relation of Exp and Loga. logb(expb(x)) = x or logb(bx) = x bb. expb(logb(x)) = x or blog (x) = xExample B: Simplifya. log2(2–5) = –5b. 8log 8(xy) =c. e2 + ln(7) =
- 63. Properties of LogarithmSo for all pairs of expb(x) & logb(x) and an input x x expb (x) # logb(#) x x (x > 0) logb (x) # expb(#) xHere is the inverse relation stated in function notation.The Inverse Relation of Exp and Loga. logb(expb(x)) = x or logb(bx) = x bb. expb(logb(x)) = x or blog (x) = xExample B: Simplifya. log2(2–5) = –5b. 8log 8(xy) = xyc. e2 + ln(7) =
- 64. Properties of LogarithmSo for all pairs of expb(x) & logb(x) and an input x x expb (x) # logb(#) x x (x > 0) logb (x) # expb(#) xHere is the inverse relation stated in function notation.The Inverse Relation of Exp and Loga. logb(expb(x)) = x or logb(bx) = x bb. expb(logb(x)) = x or blog (x) = xExample B: Simplifya. log2(2–5) = –5b. 8log 8(xy) = xyc. e2 + ln(7) = e2·eln(7)
- 65. Properties of LogarithmSo for all pairs of expb(x) & logb(x) and an input x x expb (x) # logb(#) x x (x > 0) logb (x) # expb(#) xHere is the inverse relation stated in function notation.The Inverse Relation of Exp and Loga. logb(expb(x)) = x or logb(bx) = x bb. expb(logb(x)) = x or blog (x) = xExample B: Simplifya. log2(2–5) = –5b. 8log 8(xy) = xyc. e2 + ln(7) = e2·eln(7)
- 66. Properties of LogarithmSo for all pairs of expb(x) & logb(x) and an input x x expb (x) # logb(#) x x (x > 0) logb (x) # expb(#) xHere is the inverse relation stated in function notation.The Inverse Relation of Exp and Loga. logb(expb(x)) = x or logb(bx) = x bb. expb(logb(x)) = x or blog (x) = xExample B: Simplifya. log2(2–5) = –5b. 8log 8(xy) = xyc. e2 + ln(7) = e2·eln(7) = 7e2

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