The Properties of Logarithms — Transcript

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1. 0:00Welcome to a lesson on the properties of  logarithms. Let's go ahead and get started.
2. 0:05The first property is log base a of one  is equal to zero. This is true because
3. 0:11a to the power of zero will always equal one. And  if we want an example of this, we could say log
4. 0:20one of any base, let's just say eight, will always  equal zero because a to the zero is equal to one.
5. 0:26Property two says log base a of a is equal to one
6. 0:30since a to the power of one will equal a.  So if the base and the number are the same,
7. 0:36it'll always equal one. So an example of that  might be log base five of five will equal one.
8. 0:45And then number three, log base a of a to the  power of x equals x. Well, that's true because
9. 0:52if I take a and raise it to the power of x,
10. 0:56it will equal a to the x. But the pattern we see  here is if the base and this base are the same,
11. 1:02it'll always just equal the exponent here. So  an example of this might be log base two of two
12. 1:11to the seventh will just equal seven because  two to the seventh is equal to the seventh.
13. 1:17Okay, the three main properties we want to look  at in this video are the product, quotient,
14. 1:21and power properties of logarithms. So let's go  ahead and take a look at those one at a time.
15. 1:26The product property of logarithms  states that the log of a product
16. 1:29equals the sum of the logs. So if we have log base  a of u times v, this is equal to log base a of u
17. 1:37plus log base a of v. And we can also  state the same rule using natural logs.
18. 1:43Now, since logarithms are exponents, there's  a connection between the product property for
19. 1:48exponents and the product property for logarithms.  Notice here when you're multiplying and the bases
20. 1:54are the same, you add your exponents. Well, here,  when you're multiplying your numbers, you add your
21. 1:59logarithms. But remember, the logarithms  are exponents, so if you're multiplying,
22. 2:03you add for both the product property of  logarithms and the product property for exponents.
23. 2:09So for example, if we had log base  five of six, we know we can write six
24. 2:16as two times three, which means you can  rewrite this as log base five of two plus
25. 2:24log base five of three using the  product property of logarithms.
26. 2:30Now, the quotient property of logarithms  states the log of a quotient equals the
27. 2:35difference of the logs. So if we have log base  a of u divided by v, we can rewrite this as log
28. 2:41base a of u minus log base a of v. And again,  we can write this using natural logs as well.
29. 2:47Again, making the connection to  the quotient property of exponents,
30. 2:52if you're dividing and the bases are the same, you  subtract your exponents. So when you're dividing
31. 2:57the numbers in the log, you would subtract your  logarithms. Remember, logarithms are exponents.
32. 3:03So for example, if we had log base seven  of, let's say, one point five, well,
33. 3:08one point five is the same as three halves. We  could rewrite this as log base seven of three
34. 3:17minus log base seven of two using  the quotient property of logarithms.
35. 3:22If you have a quotient, you can  rewrite the logs as a difference.
36. 3:26And the last property is the power property of  logarithms, and it states that the log of a power
37. 3:32equals the product of the power and the log.  So log base a of u to the n power is equal to
38. 3:39n times log base a of u. And again, we  can write this using natural logs as well.
39. 3:46The connection you can make to the power  property of exponents is if a power to a power,
40. 3:52you multiply your exponents. Well here, if we have  a log of a power, then we multiply n times the
41. 3:58log. Well, n is an exponent, and the log is also  an exponent, so we're multiplying our exponents.
42. 4:04An example of this one might be if we have  log base twelve of five to the eleventh power,
43. 4:11that's just equal to eleven  times log base twelve of five.
44. 4:18There are usually two types of problems that  you're asked to do to illustrate the properties
45. 4:22of logs. The first type is to expand a log as  much as possible using the properties of logs.
46. 4:29So on this problem, the first thing we should  notice is there's no rule here that deals with
47. 4:34square roots or radicals, so let's  rewrite this using rational exponents.
48. 4:38So this is equal to log base three of xy  cubed over z to the power of one half.
49. 4:45Now we'll take this one step at a time.  If we wanted to eliminate the fraction,
50. 4:49we could rewrite this as a difference of two logs.
51. 4:52It'll be the log of the numerator minus the log  of the denominator. Let's go ahead and do that.
52. 4:57So we'd have log base three of the numerator xy  cubed minus log base three of z to the one half.
53. 5:06Let's take a look at this first  log. Now we have a product,
54. 5:09so what we can do is rewrite the log  of the product as the sum of two logs.
55. 5:16So this would be equal to log base three of x plus
56. 5:22log base three of y cubed, and then minus  log base three of z to the one half.
57. 5:27Now we can apply the power property  of logs, which says we can take this
58. 5:31exponent over to the front as a  product and the same thing here.
59. 5:36So this is expanded as much as possible  as log base three of x plus three
60. 5:43times log base three of y minus  one half log base three of z.
61. 5:51And this is considered as expanded as  possible. So we're not really solving here;
62. 5:55we're just demonstrating that we  understand these three properties.
63. 5:59Let's go and take a look at  the second type of problem
64. 6:01where we're given a sum or difference of  logs, and we want to write it as a single log
65. 6:06in order to utilize the product and  quotient property of logarithms.
66. 6:10The coefficients of the logs have to be one,
67. 6:14so we first have to utilize the power property of  logs in the opposite direction that we just did.
68. 6:19We're going to take the coefficient of the log  and move it to the position of the exponent.
69. 6:24So let's go ahead and do that first.
70. 6:26We're going to move this two, so  it's natural log of x squared plus
71. 6:32now the number here is x plus three, so we'll take  this one third and move it to the position of the
72. 6:37exponent, so we'll have natural log of x plus  three to the one third power minus, take this
73. 6:44four, move it to the position of the exponent,  so we'll have natural log of two x to the fourth.
74. 6:50Now let's go ahead and take this  one step at a time. We can combine
75. 6:54these two since it's a sum of two logs;  we can multiply the number parts together.
76. 7:00So this would be natural log of x squared  times x plus three to the power of one
77. 7:07third minus natural log of two x to the fourth.
78. 7:12Now we have a difference of two logs, so we  can combine those by writing it as a quotient.
79. 7:18So we'll have the natural log,  and then in the numerator,
80. 7:20we're going to have x squared times  x plus three to the one third power,
81. 7:24and our denominator will be two  x raised to the power of four.
82. 7:28Now there is one more thing that  they might try to do to us here. Two
83. 7:31x to the power of four is  actually sixteen x to the fourth.
84. 7:36Let's take a look at this  expression inside the log.
85. 7:39We have x squared times x plus  three to the power of one third,
86. 7:44and again our denominator would be two x to the  power of four. That'd be sixteen x to the fourth.
87. 7:50We could simplify the x squared  and the x to the fourth.
88. 7:52This would simplify out, and  this would become x squared.
89. 7:56So let's go ahead and rewrite this one more time.  So we'd have the natural log of x plus three to
90. 8:03the one third power in our numerator, and our  denominator would just be sixteen x squared.
91. 8:09So after combining your logs, if you can  simplify this expression, you should if possible.
92. 8:16Okay, that'll do it for this video. I  hope you found it helpful. Thank you.