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Ex 1: Evaluate a Natural Logarithmic Expression Using the Properties of Logarithms — Transcript

by Mathispower4u · 792 words · 91 segments · language en · Watch on YouTube
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  1. 0:00- WE WANT TO EVALUATE THE LOGARITHMS
  2. 0:02GIVEN NATURAL LOG X = 3, NATURAL LOG Y = 4,
  3. 0:06AND NATURAL LOG Z = 5.
  4. 0:09SO TO EVALUATE THESE,
  5. 0:10WE'LL EXPAND THESE LOGARITHMS AS MUCH AS POSSIBLE
  6. 0:13USING THE PROPERTIES OF LOGARITHMS
  7. 0:15AND THEN PERFORM SUBSTITUTION
  8. 0:17FOR NATURAL LOG X, NATURAL LOG Y, AND NATURAL LOG Z.
  9. 0:21SO LOOKING AT THE FIRST LOGARITHM,
  10. 0:23NOTICE HOW WE HAVE NATURAL LOG OF A QUOTIENT
  11. 0:26AND SINCE WE HAVE THE NATURAL LOG OF A QUOTIENT
  12. 0:28WE CAN EXPAND THIS TO THE DIFFERENCE OF TWO LOGARITHMS
  13. 0:32USING THE QUOTIENT PROPERTY GIVEN HERE.
  14. 0:35WHERE WE'LL HAVE THE LOG OF THE NUMERATOR
  15. 0:37MINUS THE LOG OF THE DENOMINATOR.
  16. 0:40SO THIS IS EQUAL TO NATURAL LOG X CUBED MINUS
  17. 0:46BECAUSE OF THE QUOTIENT NATURAL LOG Y SQUARED Z.
  18. 0:51NOW LOOKING AT THE SECOND LOG,
  19. 0:53NOTICE HOW WE HAVE NATURAL LOG OF A PRODUCT
  20. 0:56WHICH WE CAN EXPAND USING THE PRODUCT PROPERTY OF LOGARITHMS
  21. 0:59GIVEN HERE WHERE LOG X x Y = LOG X + LOG Y.
  22. 1:07WE HAVE TO BE CAREFUL HERE
  23. 1:08BECAUSE IF WE'RE SUBTRACTING THIS SECOND LOGARITHM
  24. 1:10WE'LL HAVE TO SUBTRACT THE EXPANSION
  25. 1:13OR THE SUM OF THE LOGARITHMS.
  26. 1:15SO THIS IS EQUAL TO NATURAL LOG X CUBED
  27. 1:20AND THEN MINUS THE SUM OF THE TWO LOGS
  28. 1:24WHICH WOULD BE NATURAL LOG Y SQUARED + NATURAL LOG Z.
  29. 1:33IF WE LEAVE OFF THESE BRACKETS
  30. 1:35NOTICE HOW WE'D ONLY BE SUBTRACTING
  31. 1:36THE FIRST LOGARITHM WHICH WOULD BE INCORRECT.
  32. 1:39AND NOW FOR THE NEXT STEP,
  33. 1:41WE'RE GOING TO APPLY THE POWER PROPERTY OF LOGARITHMS
  34. 1:43GIVEN HERE
  35. 1:44WHERE WE CAN TAKE THE EXPONENT HERE AND HERE
  36. 1:48AND MOVE IT TO THE FRONT.
  37. 1:52SO NOW WE WOULD HAVE 3 NATURAL LOG X - 2 NATURAL LOG Y
  38. 2:02AND + NATURAL LOG Z.
  39. 2:05NOTICE IN THIS FORM,
  40. 2:07WE CAN NOW SUBSTITUTE FOR NATURAL LOG X, NATURAL LOG Y,
  41. 2:10AND NATURAL LOG Z.
  42. 2:12IF WE WANTED TO WE COULD CLEAR THESE BRACKETS
  43. 2:13BY DISTRIBUTING -1,
  44. 2:16BUT LET'S GO AHEAD AND LEAVE IT IN THIS FORM.
  45. 2:18SO WE'D HAVE 3 x NATURAL LOG X WHICH IS EQUAL TO 3, MINUS
  46. 2:24AND THEN IN BRACKETS THE QUANTITY 2X NATURAL LOG Y
  47. 2:28WHERE NATURAL LOG Y IS 4 PLUS NATURAL LOG Z WHICH IS 5.
  48. 2:35SO THIS WOULD BE 9 - THIS WOULD BE 8 + 5 OR 13.
  49. 2:41AND 9 - 13 IS EQUAL TO -4.
  50. 2:49OKAY. LET'S TAKE A LOOK AT OUR SECOND EXAMPLE.
  51. 2:53THE FIRST THING WE SHOULD DO HERE
  52. 2:54IS WRITE THE SQUARE ROOT USING A RATIONAL EXPONENT.
  53. 2:58IF WE HAVE THE SQUARE ROOT OF "A,"
  54. 3:02BECAUSE IT'S A SQUARE ROOT THE INDEX WOULD BE 2,
  55. 3:04THE EXPONENT WOULD BE 1.
  56. 3:06THIS IS EQUAL TO "A" TO THE 1/2 POWER,
  57. 3:09WHICH MEANS YOU CAN WRITE THIS
  58. 3:11AS NATURAL LOG X SQUARED Y TO THE 3rd,
  59. 3:16Z TO THE 5th TO THE 1/2 POWER
  60. 3:21AND IN THIS FORM WE HAVE POWERS RAISED TO POWERS,
  61. 3:25SO IF WE HAVE "A" TO THE M RAISED TO THE POWER OF N
  62. 3:30THE RULE IS THAT WE MULTIPLY THE EXPONENTS.
  63. 3:33SO LET'S GO AHEAD AND DO THAT.
  64. 3:37THIS IS EQUAL TO NATURAL LOG, 2 x 1/2 WOULD BE 1
  65. 3:41SO JUST X TO THE 1st, Y TO THE 3 x 1/2 OR Y TO THE 3/2 POWER
  66. 3:49AND THEN Z TO THE 5 x 1/2 WOULD BE Z TO THE 5/2 POWER.
  67. 3:55AND NOW WE CAN EXPAND THIS.
  68. 3:57NOTICE HOW WE HAVE A PRODUCT
  69. 3:59THEREFORE WE CAN WRITE THIS AS A SUM OF NOT ONLY 2 LOGS
  70. 4:02BUT 3 LOGS INVOLVING X, Y, AND Z.
  71. 4:07SO THIS IS EQUAL TO NATURAL LOG X
  72. 4:12+ NATURAL LOG Y TO THE 3/2 + NATURAL LOG Z TO THE 5/2
  73. 4:23AND THEN FINALLY WE CAN APPLY
  74. 4:25THE POWER PROPERTY OF LOGARITHMS
  75. 4:27WHICH MEANS WE CAN TAKE THE EXPONENT HERE AND HERE
  76. 4:31AND MOVE IT TO THE FRONT SO IT'S THE COEFFICIENT.
  77. 4:36SO NOW WE HAVE NATURAL LOG X + 3/2 NATURAL LOG Y
  78. 4:44AND + 5/2 NATURAL LOG Z.
  79. 4:50AND NOW WE CAN PERFORM SUBSTITUTION ONCE AGAIN
  80. 4:52FOR NATURAL LOG X, NATURAL LOG Y, AND NATURAL LOG Z.
  81. 4:58NATURAL LOG X = 3 + 3/2 x NATURAL LOG Y, WHICH IS 4,
  82. 5:07WHICH I'LL WRITE AS 4/1 + 5/2 x NATURAL LOG Z
  83. 5:15WHERE NATURAL LOG Z = 5, WHICH I'LL WRITE AS 5/1.
  84. 5:21NOTICE HOW THIS PRODUCT HERE WILL SIMPLIFY.
  85. 5:24THERE'S ONE FACTOR OF 2 IN 2 AND 2 FACTORS OF 2 IN 4.
  86. 5:29NOTHING SIMPLIFIES HERE SO WE HAVE 3 + THIS WOULD BE 6,
  87. 5:38+ THIS WOULD BE 25/2 OR 12.5.
  88. 5:46SO 3 + 6 IS 9 AND 9 = 18/2.
  89. 5:5118/2 + 25/2 IS EQUAL TO 43/2 OR IF WE WANT 21.5.
  90. 6:14THAT'S GOING TO DO IT FOR THESE TWO EXAMPLES.
  91. 6:16I HOPE YOU FOUND THIS HELPFUL.

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