Ex 1: Expand Logarithmic Expressions — Transcript

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1. 0:01- SINCE LOGARITHMS ARE EXPONENTS
2. 0:03AND THERE ARE RULES FOR EXPONENTS
3. 0:05THERE ARE CORRESPONDING RULES FOR LOGARITHMS
4. 0:07WHICH WE SEE HERE BELOW IN RED
5. 0:09THAT CAN BE USED TO EXPAND OR CONDENSE LOGARITHMS.
6. 0:13FOR THE PRODUCT PROPERTY OF LOGARITHMS
7. 0:15IF WE HAVE THE LOG OF X x Y
8. 0:18WE CAN WRITE THIS AS THE SUM OF 2 LOGS
9. 0:20WHERE WE HAVE LOG X + LOG Y.
10. 0:25IF WE'RE TAKING THE LOG OF A QUOTIENT
11. 0:27WE CAN WRITE IT AS A DIFFERENCE OF 2 LOGS.
12. 0:29SO THE LOG OF X DIVIDED BY Y IS EQUAL TO LOG X - LOG Y.
13. 0:36SO IF WE HAVE A QUOTIENT WE HAVE A DIFFERENCE OF 2 LOGS.
14. 0:39IF WE HAVE A PRODUCT WE HAVE A SUM OF 2 LOGS.
15. 0:42AND THEN FOR THE POWER PROPERTY OF LOGARITHMS
16. 0:44IF WE HAVE LOG OF X TO THE nth POWER,
17. 0:49WE CAN WRITE THIS AS N x LOG X.
18. 0:54SO IF WANT TO EXPAND LOG BASE 2 OF X TO THE 4th
19. 0:58x THE SQUARE ROOT OF Y TO THE 5th,
20. 1:01WE NEED RECOGNIZE THIS FIRST AS A PRODUCT
21. 1:03WHICH MEANS WE CAN WRITE THIS
22. 1:04AS A SUM OF 2 DIFFERENT LOGARITHMS.
23. 1:07BUT BEFORE WE DO THIS
24. 1:09WE DO WANT TO REWRITE THE SQUARE ROOT OF X
25. 1:10TO THE 5th IN EXPONENT FORM
26. 1:13AND SINCE WE HAVE A SQUARE ROOT WHERE THE INDEX IS 2
27. 1:16WE CAN REWRITE THIS AS LOG BASE 2 OF X TO THE 4th
28. 1:23x THIS WOULD BE Y TO THE 5/2.
29. 1:29AND AGAIN, NOW BECAUSE WE HAVE A PRODUCT
30. 1:32WE CAN WRITE THIS AS A SUM OF 2 LOGS
31. 1:35SO WE'D HAVE LOG BASE 2 OF X TO THE 4th +
32. 1:40AGAIN BECAUSE WE HAVE A PRODUCT,
33. 1:43LOG BASE 2 OF Y TO THE 5/2.
34. 1:47NOW TO EXPAND THIS ONE MORE TIME
35. 1:49WE CAN APPLY THE POWER PROPERTY
36. 1:51OF LOGARITHMS GIVEN HERE
37. 1:53WHICH MEANS WE CAN TAKE THE EXPONENTS
38. 1:55AND MOVE THEM TO THE FRONT SO THEY'RE COEFFICIENTS.
39. 1:58SO WE CAN MOVE THIS 4 HERE
40. 2:00TO THE POSITION OF THE COEFFICIENT IN FRONT
41. 2:02AND WE CAN MOVE THIS 5/2 TO THE FRONT
42. 2:05SO IT'S ALSO THE COEFFICIENT.
43. 2:07SO WE'D HAVE 4 LOG BASE 2 OF X + 5/2 x LOG BASE 2 OF Y.
44. 2:17THIS WOULD BE CONSIDERED THE EXPANDED FORM
45. 2:19OF THE GIVEN LOGARITHM.
46. 2:24LOOKING AT OUR SECOND EXAMPLE,
47. 2:25WE HAVE NATURAL LOG OF X TO THE 3rd x Y TO THE 2nd
48. 2:29DIVIDED BY Z TO THE 5th.
49. 2:31SO WE'LL DO THIS IN STAGES.
50. 2:33LET'S FIRST DEAL WITH THIS QUOTIENT HERE.
51. 2:35REMEMBER IF WE HAVE THE LOG OF A QUOTIENT
52. 2:37WE CAN WRITE IT AS A DIFFERENCE OF 2 LOGS.
53. 2:39THIS IS EQUAL TO THE NATURAL LOG OF THE NUMERATOR
54. 2:43WHICH IS X TO THE 3rd, Y TO THE 2nd -
55. 2:47BECAUSE WE HAVE A QUOTIENT, NATURAL LOG Z TO THE 5th.
56. 2:51AND NOW LOOKING AT THE FIRST LOG
57. 2:53BECAUSE WE STILL HAVE A PRODUCT HERE
58. 2:55WE CAN WRITE THIS LOG
59. 2:56OR EXPAND THIS LOG INTO A SUM OF 2 LOGS.
60. 3:00WE'D HAVE NATURAL LOG X TO THE 3rd + NATURAL LOG Y TO THE 2nd
61. 3:07AND THEN WE STILL HAVE - NATURAL LOG Z TO THE 5th.
62. 3:11AND NOW WE CAN APPLY THE POWER PROPERTY OF LOGARITHMS
63. 3:13WHICH MEANS WE CAN MOVE THESE EXPONENTS
64. 3:15TO THE FRONT OF THE LOGARITHMS.
65. 3:22SO TO EXPAND THIS FURTHER WE HAVE 3 NATURAL LOG X
66. 3:26+ THIS WOULD BE 2 NATURAL LOG Y - 5 NATURAL LOG Z.
67. 3:36AND AGAIN, THIS WOULD BE CONSIDERED THE EXPANDED FORM
68. 3:38OF THE GIVEN LOGARITHM.
69. 3:44OKAY. WE'LL TAKE A LOOK AT SOME MORE EXAMPLES
70. 3:46IN THE NEXT VIDEO.