1 00:00:00,422 --> 00:00:02,512 Let's learn a little bit about the wonderful 2 00:00:02,512 --> 00:00:06,209 world of logarithms. 3 00:00:06,209 --> 00:00:09,519 So we already know how to take exponents. 4 00:00:09,519 --> 00:00:13,871 If I were to say 2 to the fourth power, what does that mean? 5 00:00:13,871 --> 00:00:18,846 Well that means 2 times 2 times 2 times 2. 6 00:00:18,846 --> 00:00:24,349 2 multiplied or repeatedly multiplied 4 times, and so 7 00:00:24,349 --> 00:00:26,499 this is going to be 8 00:00:26,499 --> 00:00:27,630 2 times 2 is 4 9 00:00:27,630 --> 00:00:32,887 times 2 is 8, times 2 is 16. 10 00:00:32,887 --> 00:00:35,346 But what if we think about things in another way. 11 00:00:35,346 --> 00:00:40,496 We know that we get to 16 when we raise 2 to some power 12 00:00:40,496 --> 00:00:42,908 but we want to know what that power is. 13 00:00:42,908 --> 00:00:46,651 So for example, let's say that I start with 2, and I say 14 00:00:46,651 --> 00:00:52,832 I'm raising it to some power, what does that power have to be to get 16? 15 00:00:52,832 --> 00:01:00,369 Well we just figured that out. 'X' would have to be 4. 16 00:01:00,369 --> 00:01:04,762 And this is what logarithms are fundamentally about, figuring out what power 17 00:01:04,762 --> 00:01:08,514 you have to raise to, to get another number. 18 00:01:08,514 --> 00:01:12,288 Now the way that we would denote this with logarithm notation 19 00:01:12,288 --> 00:01:15,269 is we would say, log, base-- 20 00:01:15,269 --> 00:01:18,182 actually let me make it a little bit more colourful. 21 00:01:18,182 --> 00:01:21,656 Log, base 2-- 22 00:01:21,656 --> 00:01:23,592 I'll do this 2 in blue... 23 00:01:23,592 --> 00:01:31,777 Log, base 2, of 16 24 00:01:31,777 --> 00:01:36,118 is equal to what, or is equal in this case since we have the 'x' there, 25 00:01:36,118 --> 00:01:39,975 is equal to 'x'? 26 00:01:39,975 --> 00:01:45,063 This and this are completely equivalent statements. 27 00:01:45,063 --> 00:01:49,078 This is saying "hey well if I take 2 to some 'x' power I get 16'." 28 00:01:49,078 --> 00:01:57,388 This is saying, "what power do I need to raise 2 to, to get 16 and I'm going to set that to be equal to 'x'." 29 00:01:57,388 --> 00:01:59,927 And you'll say, "well you have to raise it to the fourth power and once again 30 00:01:59,927 --> 00:02:01,583 'x' is equal to 4. 31 00:02:01,583 --> 00:02:07,670 So with that out of the way let's try more examples of evaluating logarithmic expressions. 32 00:02:07,670 --> 00:02:12,051 Let's say you had... 33 00:02:12,051 --> 00:02:25,127 log, base 3, of 81. What would this evaluate to? 34 00:02:25,127 --> 00:02:31,319 Well this is a reminder, this evaluates to the power we have to raise 3 to, to get 35 00:02:31,319 --> 00:02:33,053 to 81. 36 00:02:33,053 --> 00:02:37,764 So if you want to, you can set this to be equal to an 'x', and you can 37 00:02:37,764 --> 00:02:48,779 restate this equation as, 3 to the 'x' power, is equal to 81. 38 00:02:48,779 --> 00:02:52,287 Why is a logarithm useful? And you'll see that it has very interesting properties 39 00:02:52,287 --> 00:02:53,112 later on. 40 00:02:53,112 --> 00:02:57,797 But you didn't necessarily have to use algebra. To do it this way, to say that 'x' is the power 41 00:02:57,797 --> 00:03:02,151 you raise 3 to to get to 81, you had to use algebra here, while with just 42 00:03:02,151 --> 00:03:06,257 a straight up logarithmic expression, you didn't really have to use any algebra, we didn't have to 43 00:03:06,257 --> 00:03:09,248 say that it was equal to 'x', we could just say that this evaluates to 44 00:03:09,248 --> 00:03:12,614 the power I need to raise 3 to to get to 81. 45 00:03:12,614 --> 00:03:15,895 The power I need to raise 3 to to get to 81. 46 00:03:15,895 --> 00:03:19,398 Well what power do you have to raise 3 to to get to 81? 47 00:03:19,398 --> 00:03:21,244 Well let's experiment a little bit, 48 00:03:21,244 --> 00:03:25,799 so 3 to the first power is just 3, 3 to the second power is 9, 49 00:03:25,799 --> 00:03:34,383 3 to the third power is 27, 3 to the fourth power, 27 times 3 is equal to 81. 50 00:03:34,383 --> 00:03:36,967 3 to the fourth power is equal to 81. 51 00:03:36,967 --> 00:03:39,388 'X' is equal to four. 52 00:03:39,388 --> 00:03:41,169 So we could say... 53 00:03:41,169 --> 00:03:50,953 Log, base 3, of 81, is equal to-- 54 00:03:50,953 --> 00:03:53,993 I'll do this in a different colour. 55 00:03:53,993 --> 00:03:57,453 Is equal to 4. 56 00:03:57,453 --> 00:04:00,730 Let's do several more of these examples and I really encourage you to give a shot 57 00:04:00,730 --> 00:04:04,548 on your own and hopefully you'll get the hang of it. 58 00:04:04,548 --> 00:04:09,647 So let's try a larger number, let's say we want to take 59 00:04:09,647 --> 00:04:17,511 log, base 6, of 216. What will this evaluate to? 60 00:04:17,511 --> 00:04:22,768 Well we're asking ourselves, "what power do we have to raise 6 to, to get 216?" 61 00:04:22,768 --> 00:04:27,237 6 to the first power is 6, 6 to the second power is 36, 62 00:04:27,237 --> 00:04:31,411 36 times 6 is 216. 63 00:04:31,411 --> 00:04:34,330 This is equal to 216. 64 00:04:34,330 --> 00:04:37,451 So this is 6 to the third power is equal to 216. 65 00:04:37,451 --> 00:04:40,637 So if someone says "what power do I have to raise 6 to-- 66 00:04:40,637 --> 00:04:41,851 this base here, 67 00:04:41,851 --> 00:04:44,328 to get to 216?" 68 00:04:44,328 --> 00:04:48,096 Well that's just going to be equal to 3. 69 00:04:48,096 --> 00:04:52,063 6 to the third power is equal to 216. 70 00:04:52,063 --> 00:04:56,891 Let's try another one. Let's say I had, I dunno, 71 00:04:56,891 --> 00:05:02,599 log, base 2, of 64. 72 00:05:02,599 --> 00:05:06,056 So what does this evaluate to? 73 00:05:06,056 --> 00:05:11,749 Well once again we're asking ourselves, "well this will evaluate to the exponent that I have to raise this 74 00:05:11,749 --> 00:05:14,545 base 2, and you do this as a little subscript right here. 75 00:05:14,545 --> 00:05:18,748 The exponent that I have to raise 2 to, to get to 64." 76 00:05:18,748 --> 00:05:22,217 So 2 to the first power is 2, 2 to the second power is 4, 77 00:05:22,217 --> 00:05:27,369 8, 16, 32, 64. 78 00:05:27,369 --> 00:05:34,663 So this right over here is 2 to the sixth power, is equal to 64. So when you evaluate this expression 79 00:05:34,663 --> 00:05:38,518 you say "what power do I have to raise 2 to, to get to 64?" 80 00:05:38,518 --> 00:05:43,812 Well I have to raise to the sixth power. 81 00:05:43,812 --> 00:05:49,097 Let's do a slightly more straightforward one, or maybe this will be less 82 00:05:49,097 --> 00:05:51,349 straightforward depending on how you view it. 83 00:05:51,349 --> 00:05:58,245 What is log, base 100, of 1? 84 00:05:58,245 --> 00:06:00,626 I'll let you think about that for a second. 85 00:06:00,626 --> 00:06:05,101 100 is a subscript so it's, log, base 100, of 1. 86 00:06:05,101 --> 00:06:08,091 That's one way to think about it, I'll put parentheses around the 1. 87 00:06:08,091 --> 00:06:10,991 What does this evaluate to? 88 00:06:10,991 --> 00:06:15,044 Well this is asking ourselves, or we would evaluate this as, 89 00:06:15,044 --> 00:06:22,197 "what power do I have to raise 100 to, to get to 1?" 90 00:06:22,197 --> 00:06:24,241 So let me write this down as an equation. 91 00:06:24,241 --> 00:06:26,909 If I set this to be equal to 'x', this is literally saying 92 00:06:26,909 --> 00:06:31,598 100, to what power, is equal to 1? 93 00:06:31,598 --> 00:06:34,424 Well anything that a 0 power is equal to 1. 94 00:06:34,424 --> 00:06:36,707 So in this case 'x' is equal to 0. 95 00:06:36,707 --> 00:06:44,114 So log, base 100, of 1, is going to be equal to 0. 96 00:06:44,114 --> 00:06:50,933 Log base anything of 1, is going to be equal to 0 because anything to the 0 97 00:06:50,933 --> 00:06:55,905 power and we're not talking about 0 here. Anything that is to the power of 0 98 00:06:55,905 --> 00:07:01,328 that is not 0, is going to be equal to 1.