1 00:00:00,426 --> 00:00:02,625 Let's give ourselves a little more practice 2 00:00:02,625 --> 00:00:04,246 with logarithms. 3 00:00:04,246 --> 00:00:05,975 So just a little bit of review... 4 00:00:05,975 --> 00:00:11,474 Let's evaluate log, base 2, of 8. 5 00:00:11,474 --> 00:00:14,693 What does this evaluate to? 6 00:00:14,693 --> 00:00:19,780 Well it's asking us it'll evaluate to the power that I have 7 00:00:19,780 --> 00:00:23,449 to raise our base to. That I have to raise 2 to, 8 00:00:23,449 --> 00:00:25,249 to get to 8. 9 00:00:25,249 --> 00:00:27,983 So 2 to the first power is 2, 2 to the second power is 4, 10 00:00:27,983 --> 00:00:32,933 2 to the third power is 8. So this right over here is going to 11 00:00:32,933 --> 00:00:34,395 be equal to 3. 12 00:00:34,395 --> 00:00:37,383 Fair enough, we did examples like that in the last video. 13 00:00:37,383 --> 00:00:39,383 Let's do something a little bit more interesting... 14 00:00:39,383 --> 00:00:41,815 What is-- 15 00:00:41,815 --> 00:00:43,128 and I'll colour code it... 16 00:00:43,128 --> 00:00:51,128 what is log, base 8, of 2? 17 00:00:51,128 --> 00:00:54,427 Now this is interesting, I'll give you a few seconds to 18 00:00:54,427 --> 00:00:56,378 think about it. 19 00:00:56,378 --> 00:00:59,377 Well we're asking ourselves or this will evaluate to 20 00:00:59,377 --> 00:01:04,028 the exponent that I have to raise 8 to, to get to 2. 21 00:01:04,028 --> 00:01:06,078 So let's think about that in another way. 22 00:01:06,078 --> 00:01:12,828 So we could say 8, to some power, and that exponent that I'm 23 00:01:12,828 --> 00:01:16,112 raising 8 to is essentially what this logarithm would evaluate to. 24 00:01:16,112 --> 00:01:24,643 8 to some power, is going to be equal to 2. 25 00:01:24,643 --> 00:01:28,472 Well if 2 to the third power is 8, 8 to the one-third power 26 00:01:28,472 --> 00:01:30,113 is equal to 2. 27 00:01:30,113 --> 00:01:32,964 So 'x' is equal to one-third. 28 00:01:32,964 --> 00:01:35,596 8 to the one-third power is equal to 2. 29 00:01:35,596 --> 00:01:38,781 Or you could say that the cube root of 8 is 2. 30 00:01:38,781 --> 00:01:40,914 So in this case 'x' is one-third. 31 00:01:40,914 --> 00:01:47,313 This logarithm right over here will evaluate to one-third. 32 00:01:47,313 --> 00:01:48,652 Fascinating. 33 00:01:48,652 --> 00:01:50,911 Let's mix it up a little bit more. 34 00:01:50,911 --> 00:01:56,314 Let's say we have the log, base 2, instead of 8 35 00:01:56,314 --> 00:02:01,431 let's put a one-eighth, right over here. 36 00:02:01,431 --> 00:02:05,163 So I'll give you a few seconds to think about that. 37 00:02:05,163 --> 00:02:08,762 Well it's asking us or this will evaluate to the exponent that I have to 38 00:02:08,762 --> 00:02:11,564 raise 2 to, to get to one-eighth. 39 00:02:11,564 --> 00:02:15,495 So if we set this to be equal to 'x', we're essentially saying 40 00:02:15,495 --> 00:02:25,335 2 to the 'x' power, is equal to one-eighth. 41 00:02:25,335 --> 00:02:27,082 Well we know that 2 to the third power-- 42 00:02:27,082 --> 00:02:28,465 Let me write this down... 43 00:02:28,465 --> 00:02:31,715 We already know that 2 to the third power is equal to 8. 44 00:02:31,715 --> 00:02:35,147 If we want to get to one-eighth, which is a reciprocal of 8. 45 00:02:35,147 --> 00:02:37,614 We just have to raise to the negative 3 power. 46 00:02:37,614 --> 00:02:42,798 2 to the negative 3 power is one over 2 to the third power. 47 00:02:42,798 --> 00:02:46,331 Which is the same thing as one over eight. 48 00:02:46,331 --> 00:02:49,514 So, if we're asking ourselves "what exponent do we have to raise 49 00:02:49,514 --> 00:02:51,765 2 to to get to one-eighth?" 50 00:02:51,765 --> 00:02:54,515 Well we have to raise it to the negative 3 power. 51 00:02:54,515 --> 00:02:56,915 So 'x' is equal to negative 3. 52 00:02:56,915 --> 00:03:01,865 This logarithm evaluates to negative 3. 53 00:03:01,865 --> 00:03:05,736 Now let's really, really mix it up. 54 00:03:05,736 --> 00:03:18,266 What would be the log, base 8, of one-half. 55 00:03:18,266 --> 00:03:21,015 What does this evaluate to? 56 00:03:21,015 --> 00:03:26,282 Let me clean this up so we have some space to work with. 57 00:03:26,282 --> 00:03:32,451 So as always, we're saying "what power do I have to raise 8 to, to get to one-half?" 58 00:03:32,451 --> 00:03:34,498 So let's think about that a little bit. 59 00:03:34,498 --> 00:03:37,433 We already know that 8 to the one-third power is 60 00:03:37,433 --> 00:03:39,015 equal to 2. 61 00:03:39,015 --> 00:03:42,701 If we want the reciprocal of 2 right over here, we have to just 62 00:03:42,701 --> 00:03:45,634 raise 8 to the negative one-third. 63 00:03:45,634 --> 00:03:51,383 So let me write that down, 8, to the negative one-third power, 64 00:03:51,383 --> 00:03:56,517 is going to be equal to one over eight, to the one-third power, 65 00:03:56,517 --> 00:03:59,400 and we already know the cube root of 8 or 8 to the one-third power is 66 00:03:59,400 --> 00:04:00,683 equal to 2. This is equal to 67 00:04:00,683 --> 00:04:02,265 one-half. 68 00:04:02,265 --> 00:04:08,001 So, the log, base 8, of one-half is equal to? 69 00:04:08,001 --> 00:04:10,551 Well the power I have to raise 8 to to get to one-half is 70 00:04:10,551 --> 00:04:15,421 negative one-third. 71 00:04:15,421 --> 00:04:19,024 I hope you enjoyed that as much as I did.