Introduction to Logarithms — Transcript

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1. 0:00Welcome to an introduction to logarithms.
2. 0:02The goals of this video are to define a logarithm
3. 0:05and convert exponential equations to  logarithmic equations and vice versa.
4. 0:11So the first thing I want to talk  about is, what is a logarithm?
5. 0:14Well, a logarithm is just an exponent.
6. 0:16So if someone asks you that,  now you have the answer.
7. 0:20Let's take a look at a more formal definition.
8. 0:23Log base a of x equals y means a  to the power of y is equal to x.
9. 0:30So since a logarithm is an exponent,
10. 0:32it should seem logical that we can write  a log equation as an exponential equation.
11. 0:37Notice a would be the base, y would be  the exponent, and x would be the number.
12. 0:45We could say that y is the  power which we raise a to get x,
13. 0:50and we should note that a has to be  greater than zero and a cannot equal one.
14. 0:56So if we have log base two of eight equals three,  this means two to the power of three equals eight.
15. 1:09And a nice way to remember that is just start  with the base and kind of go around the circle.
16. 1:13Two to the power of three is equal to eight.
17. 1:17And we can use this definition in  the opposite direction as well.
18. 1:21If we have five to the second equals twenty-five,  we can rewrite this as a log equation where we'd
19. 1:28have log base five of twenty-five must equal  two, since five to the second equals twenty-five.
20. 1:41We'll come back to this idea later.
21. 1:43Let's go ahead and talk about  common log and natural log,
22. 1:46because these are the two logs that  you'll find on your calculator.
23. 1:50The logarithm with base ten  is called the common log.
24. 1:54This is the log key on your calculator, so  you won't see common log written like this.
25. 1:59If it's base ten, we leave the base off.
26. 2:03So if you have your calculator, you'll  notice a log button here on the left side.
27. 2:07So if I press log ten thousand,
28. 2:14this is equal to four.
29. 2:16Let's go ahead and write that down.
30. 2:19And if I ask you why is log  ten thousand equal to four,
31. 2:24you should be able to tell me that, oh this  is common log, which means that's base ten,
32. 2:29and so what this means is ten to the  power of four must equal ten thousand.
33. 2:37So it's great to use the  calculator to evaluate common log,
34. 2:40but I think it's important that  you understand what you're finding.
35. 2:43Notice you're finding the exponent for which  you raise ten to to get the number ten thousand.
36. 2:50The second log the calculator  has is the natural logarithm.
37. 2:54The logarithm with base e is called  the natural logarithmic function.
38. 2:59This log is also on your  calculator as capital L, capital N.
39. 3:04So you won't see log base  e of x; it's written ln x,
40. 3:09so you have to remember that  this just means log base e.
41. 3:13Remember, e is an irrational number similar to  pi, except it's approximately equal to 2.718.
42. 3:20So for example, if we type  in natural log of ninety-five
43. 3:24on the calculator, here's the natural log key.
44. 3:28We're going to get a decimal approximation.
45. 3:30It's approximately four  point five five three nine.
46. 3:36So what does that mean?
47. 3:38Remember natural log means log base  e, so that means e to the power of
48. 3:42four point five five three nine is  approximately equal to ninety-five.
49. 3:46We did round this; we won't be exact.
50. 3:48But let's go ahead and check it.
51. 3:50So if we press second natural  log, it brings up base e.
52. 3:54We can just type in our exponent, and you  can see it's approximately ninety-five.
53. 4:01Let's go and take a look at a few more examples.
54. 4:04Let's take these logarithmic equations  and write them as exponential equations,
55. 4:08which means we're just going to rewrite  these using the definition of a logarithm.
56. 4:14So here's the definition, but  if you can just remember to
57. 4:17go clockwise around the equation,  two to the third must equal eight.
58. 4:23The nice thing about going from log form  to exponential form, you can check it;
59. 4:27this is true, therefore it's correct.
60. 4:29Here we'd have three to the power  of negative two equals one-ninth;
61. 4:37that's correct.
62. 4:40Here's natural log, so that would be base e.
63. 4:42It might be helpful to write that in e to  the first power must equal e; that's correct.
64. 4:50And then another common log
65. 4:52recognizing this as base ten, ten to the  power of three must equal a thousand.
66. 5:01And let's go ahead and take a look at a  couple more in the opposite direction.
67. 5:04We want to write these exponential  equations as logarithmic equations.
68. 5:08So on this first problem, we know  we're going to have a logarithm,
69. 5:14so there's three key components: the  base, the exponent, and the number.
70. 5:18Well, the base is four down there in the basement;  you can think of it being in a lower position.
71. 5:26A logarithm is an exponent,  so the exponent is three.
72. 5:29The log must equal three,  and the number is sixty-four.
73. 5:34Again, four to the power of three  equals sixty-four and checks.
74. 5:38The third one, again we're going to  have a logarithm here; it's base ten.
75. 5:44A logarithm is an exponent, so  the exponent is negative two.
76. 5:49The log is equal to negative two,  and then the number is one-hundredth.
77. 5:56Now there's one more thing on this  one though; it is a common log.
78. 6:00So it's not wrong to put this base ten  in there, but you'll very seldom see it.
79. 6:05So I'm going to go ahead and erase that.
80. 6:08Remember if the base isn't  there, we know it's common log.
81. 6:12And the last one, it's a  logarithm; it'll have a log.
82. 6:18Now it's base e again.
83. 6:20A logarithm is an exponent, so it must equal two,
84. 6:24and the number is approximately  seven point three eight nine.
85. 6:31Now on this one though, we have log  base e; we can't leave it in this form.
86. 6:35We should rewrite this as natural  log seven point three eight nine.
87. 6:41So this would be our intermediate step, but  we want to leave this as our final answer.
88. 6:47Okay, now you might be asking,  what's the big deal with logarithms?
89. 6:49Well, there are a lot of logarithmic  scales used in real life.
90. 6:52Three of the most common ones are  the Richter scale to measure the
91. 6:56intensity of earthquakes is a logarithmic scale.
92. 6:59A decibel scale to measure sound  levels is a logarithmic scale,
93. 7:03as well as the pH scale to measure  how basic or acidic a substance is.
94. 7:08That is also a logarithmic scale.
95. 7:10So if you search any of these online, you'll  notice that they all involve logarithms.
96. 7:15Okay, so this is the first of  several videos on logarithms.
97. 7:18I hope you found it helpful.  Thank you for watching.