1 00:00:00,000 --> 00:00:00,610 2 00:00:00,610 --> 00:00:03,585 Welcome to this presentation on logarithm properties. 3 00:00:03,585 --> 00:00:05,969 Now this is going to be a very hands-on presentation. 4 00:00:05,969 --> 00:00:08,580 If you don't believe that one of these properties are true 5 00:00:08,580 --> 00:00:11,839 and you want them proved, I've made three or four videos 6 00:00:11,839 --> 00:00:12,929 that actually prove these properties. 7 00:00:12,929 --> 00:00:14,000 But what I'm going to do is I'm going to show 8 00:00:14,000 --> 00:00:14,619 you the properties. 9 00:00:14,619 --> 00:00:15,899 And then show you how they can be used. 10 00:00:15,900 --> 00:00:17,519 It's going to be little more hands-on. 11 00:00:17,519 --> 00:00:20,769 So let's just do a little bit of a review of just 12 00:00:20,769 --> 00:00:23,000 what a logarithm is. 13 00:00:23,000 --> 00:00:28,539 So if I say that a-- Oh that's not the right. 14 00:00:28,539 --> 00:00:28,929 Let's see. 15 00:00:28,929 --> 00:00:31,949 I want to change-- There you go. 16 00:00:31,949 --> 00:00:35,259 Let's say I say that a-- Let me start over. 17 00:00:35,259 --> 00:00:41,710 a to the b is equal to c. 18 00:00:41,710 --> 00:00:44,660 So if we-- a to the b power is equal to c. 19 00:00:44,659 --> 00:00:47,419 So another way to write this exact same relationship instead 20 00:00:47,420 --> 00:00:49,969 of writing the exponent, is to write it as a logarithm. 21 00:00:49,969 --> 00:00:57,549 So we can say that the logarithm base a of 22 00:00:57,549 --> 00:01:02,039 c is equal to b. 23 00:01:02,039 --> 00:01:05,000 So these are essentially saying the same thing. 24 00:01:05,000 --> 00:01:06,750 They just have different kind of results. 25 00:01:06,750 --> 00:01:09,709 In one, you know a and b and you're kind of getting c. 26 00:01:09,709 --> 00:01:12,019 That's what exponentiation does for you. 27 00:01:12,019 --> 00:01:14,250 And the second one, you know a and you know that when 28 00:01:14,250 --> 00:01:16,340 you raise it to some power you get c. 29 00:01:16,340 --> 00:01:18,040 And then you figure out what b is. 30 00:01:18,040 --> 00:01:20,540 So they're the exact same relationship, just stated 31 00:01:20,540 --> 00:01:22,090 in a different way. 32 00:01:22,090 --> 00:01:24,909 Now I will introduce you to some interesting 33 00:01:24,909 --> 00:01:25,969 logarithm properties. 34 00:01:25,969 --> 00:01:30,480 And they actually just fall out of this relationship and 35 00:01:30,480 --> 00:01:32,740 the regular exponent rules. 36 00:01:32,739 --> 00:01:36,864 So the first is that the logarithm-- Let me do 37 00:01:36,864 --> 00:01:39,349 a more cheerful color. 38 00:01:39,349 --> 00:01:45,259 The logarithm, let's say, of any base-- So let's just call 39 00:01:45,260 --> 00:01:47,190 the base-- Let's say b for base. 40 00:01:47,189 --> 00:01:57,819 Logarithm base b of a plus logarithm base b of c-- and 41 00:01:57,819 --> 00:01:59,769 this only works if we have the same bases. 42 00:01:59,769 --> 00:02:01,799 So that's important to remember. 43 00:02:01,799 --> 00:02:12,789 That equals the logarithm of base b of a times c. 44 00:02:12,789 --> 00:02:14,659 Now what does this mean and how can we use it? 45 00:02:14,659 --> 00:02:18,319 Or let's just even try it out with some, well I 46 00:02:18,319 --> 00:02:19,629 don't know, examples. 47 00:02:19,629 --> 00:02:22,859 So this is saying that-- I'll switch to another color. 48 00:02:22,860 --> 00:02:25,390 Let's make mauve my-- Mauve-- I don't know. 49 00:02:25,389 --> 00:02:26,729 I never know how to say that properly. 50 00:02:26,729 --> 00:02:29,469 Let's make that my example color. 51 00:02:29,469 --> 00:02:40,919 So let's say logarithm of base 2 of-- I don't know --of 8 52 00:02:40,919 --> 00:02:50,569 plus logarithm base 2 of-- I don't know let's say --32. 53 00:02:50,569 --> 00:02:53,209 54 00:02:53,210 --> 00:02:58,409 So, in theory, this should equal, if we believe this 55 00:02:58,409 --> 00:03:05,500 property, this should equal logarithm base 2 of what? 56 00:03:05,500 --> 00:03:07,909 Well we say 8 times 32. 57 00:03:07,909 --> 00:03:17,659 So 8 times 32 is 240 plus 16, 256. 58 00:03:17,659 --> 00:03:18,509 Well let's see if that's true. 59 00:03:18,509 --> 00:03:20,739 Just trying out this number and this is really isn't a proof. 60 00:03:20,740 --> 00:03:22,860 But it'll give you a little bit of an intuition, I think, for 61 00:03:22,860 --> 00:03:23,990 what's going on around you. 62 00:03:23,990 --> 00:03:26,340 So log-- So this is-- We just used our property. 63 00:03:26,340 --> 00:03:28,479 This little property that I presented to you. 64 00:03:28,479 --> 00:03:29,889 And let's just see if it works out. 65 00:03:29,889 --> 00:03:31,859 So log base 2 of 8. 66 00:03:31,860 --> 00:03:34,520 2 to what power is equal to 8? 67 00:03:34,520 --> 00:03:38,960 Well 2 to the third power is equal to 8, right? 68 00:03:38,960 --> 00:03:41,310 So this term right here, that equals 3, right? 69 00:03:41,310 --> 00:03:44,810 Log base 2 of 8 is equal to 3. 70 00:03:44,810 --> 00:03:48,400 2 to what power is equal to 32? 71 00:03:48,400 --> 00:03:48,789 Let's see. 72 00:03:48,789 --> 00:03:50,579 2 to the fourth power is 16. 73 00:03:50,580 --> 00:03:53,300 2 to the fifth power is 32. 74 00:03:53,300 --> 00:03:58,410 So this right here is 2 to the-- This is 5, right? 75 00:03:58,409 --> 00:04:02,750 And 2 to the what power is equal to 256? 76 00:04:02,750 --> 00:04:05,889 Well if you're a computer science major, you'll 77 00:04:05,889 --> 00:04:07,209 know that immediately. 78 00:04:07,210 --> 00:04:10,469 That a bite can have 256 values in it. 79 00:04:10,469 --> 00:04:12,439 So it's 2 to the eighth power. 80 00:04:12,439 --> 00:04:15,520 But if you don't know that, you could multiply it out yourself. 81 00:04:15,520 --> 00:04:16,710 But this is 8. 82 00:04:16,709 --> 00:04:18,495 And I'm not doing it just because I knew that 3 83 00:04:18,495 --> 00:04:19,470 plus 5 is equal to 8. 84 00:04:19,470 --> 00:04:21,230 I'm doing this independently. 85 00:04:21,230 --> 00:04:22,300 So this is equal to 8. 86 00:04:22,300 --> 00:04:28,530 But it does turn out that 3 plus 5 is equal to 8. 87 00:04:28,529 --> 00:04:32,169 This may seem like magic to you or it may seem obvious. 88 00:04:32,170 --> 00:04:35,560 And for those of you who it might seem a little obvious, 89 00:04:35,560 --> 00:04:43,319 you're probably thinking, well 2 to the third times 2 to the 90 00:04:43,319 --> 00:04:49,509 fifth is equal to 2 to the 3 plus 5, right? 91 00:04:49,509 --> 00:04:51,800 This is just an exponent rule. 92 00:04:51,800 --> 00:04:52,670 What do they call this? 93 00:04:52,670 --> 00:04:54,750 The additive exponent prop-- I don't know. 94 00:04:54,750 --> 00:04:56,220 I don't know the names of things. 95 00:04:56,220 --> 00:04:59,920 And that equals 2 to 8, 2 to the eighth. 96 00:04:59,920 --> 00:05:03,160 And that's exactly what we did here, right? 97 00:05:03,160 --> 00:05:05,830 On this side, we had 2 the third times 2 to 98 00:05:05,829 --> 00:05:07,000 the fifth, essentially. 99 00:05:07,000 --> 00:05:09,779 And on this side, you have them added to each other. 100 00:05:09,779 --> 00:05:12,849 And what makes the logarithms interesting is and why-- It's 101 00:05:12,850 --> 00:05:13,970 a little confusing at first. 102 00:05:13,970 --> 00:05:15,620 And you can watch the proofs if you really want kind of 103 00:05:15,620 --> 00:05:18,449 a rigorous-- my proofs aren't rigorous. 104 00:05:18,449 --> 00:05:19,939 But if you want kind of a better explanation 105 00:05:19,939 --> 00:05:21,360 of how this works. 106 00:05:21,360 --> 00:05:23,480 But this should hopefully give you an tuition for why this 107 00:05:23,480 --> 00:05:25,330 property holds, right? 108 00:05:25,329 --> 00:05:27,329 Because when you multiply two numbers of the 109 00:05:27,329 --> 00:05:29,269 same base, right? 110 00:05:29,269 --> 00:05:31,719 Two exponential expressions of the same base, you 111 00:05:31,720 --> 00:05:33,560 can add their exponents. 112 00:05:33,560 --> 00:05:37,050 Similarly, when you have the log of two numbers multiplied 113 00:05:37,050 --> 00:05:41,710 by each other, that's equivalent to the log of each 114 00:05:41,709 --> 00:05:43,529 of the numbers added to each other. 115 00:05:43,529 --> 00:05:45,809 This is the same property. 116 00:05:45,810 --> 00:05:49,939 If you don't believe me, watch the proof videos. 117 00:05:49,939 --> 00:05:56,379 So let's do a-- Let me show you another log property. 118 00:05:56,379 --> 00:05:58,230 It's pretty much the same one. 119 00:05:58,230 --> 00:05:59,400 I almost view them the same. 120 00:05:59,399 --> 00:06:10,449 So this is log base b of a minus log base b of c 121 00:06:10,449 --> 00:06:17,039 is equal to log base b of-- well I ran out. 122 00:06:17,040 --> 00:06:19,290 I'm running out of space --a divided by c. 123 00:06:19,290 --> 00:06:21,540 That says a divided by c. 124 00:06:21,540 --> 00:06:25,240 And we can, once again, try it out with some numbers. 125 00:06:25,240 --> 00:06:28,620 I use 2 a lot just because 2 is an easy number to 126 00:06:28,620 --> 00:06:29,790 figure out the powers. 127 00:06:29,790 --> 00:06:30,750 But let's use a different number. 128 00:06:30,750 --> 00:06:41,199 Let's say log base 3 of-- I don't know --log base 3 of-- 129 00:06:41,199 --> 00:06:44,819 well you know, let's make it interesting --log base 3 of 130 00:06:44,819 --> 00:06:56,930 1/9 minus log base 3 of 81. 131 00:06:56,930 --> 00:07:02,500 So this property tells us-- This is the same thing as-- 132 00:07:02,500 --> 00:07:04,180 Well I'm ending up with a big number. 133 00:07:04,180 --> 00:07:12,990 Log base 3 of 1/9 divided by 81. 134 00:07:12,990 --> 00:07:15,889 So that's the same thing as 1/9 times 1/81. 135 00:07:15,889 --> 00:07:19,709 I used two large numbers for my example, but 136 00:07:19,709 --> 00:07:21,319 we'll move forward. 137 00:07:21,319 --> 00:07:21,750 So let's see. 138 00:07:21,750 --> 00:07:25,699 9 times 8 is 720, right? 139 00:07:25,699 --> 00:07:27,189 9 times-- Right. 140 00:07:27,189 --> 00:07:28,579 9 times 8 is 720. 141 00:07:28,579 --> 00:07:31,036 So this is 1/729. 142 00:07:31,036 --> 00:07:37,860 So this is log base 3 over 1/729. 143 00:07:37,860 --> 00:07:42,160 So what-- What does-- 3 to what power is equal to 1/9? 144 00:07:42,160 --> 00:07:45,340 Well 3 squared is equal to 9, right? 145 00:07:45,339 --> 00:07:48,069 146 00:07:48,069 --> 00:07:53,250 So 3-- So we know that if 3 squared is equal to 9, then we 147 00:07:53,250 --> 00:07:56,949 know that 3 to the negative 2 is equal to 1/9, right? 148 00:07:56,949 --> 00:07:58,189 The negative just inverts it. 149 00:07:58,189 --> 00:08:02,160 So this is equal to negative 2, right? 150 00:08:02,160 --> 00:08:06,120 And then minus-- 3 to what power is equal 81? 151 00:08:06,120 --> 00:08:07,870 3 to the third power is 27. 152 00:08:07,870 --> 00:08:11,389 So 3 to the fourth power. 153 00:08:11,389 --> 00:08:15,870 So we have minus 2 minus 4 is equal to-- Well, we could 154 00:08:15,870 --> 00:08:16,709 do it a couple of ways. 155 00:08:16,709 --> 00:08:20,529 Minus 2 minus 4 is equal to minus 6. 156 00:08:20,529 --> 00:08:23,279 And now we just have to confirm that 3 to the minus sixth 157 00:08:23,279 --> 00:08:26,079 power is equal to 1/729. 158 00:08:26,079 --> 00:08:26,889 So that's my question. 159 00:08:26,889 --> 00:08:34,429 Is 3 to the minus sixth power, is that equal to 7-- 1/729? 160 00:08:34,429 --> 00:08:36,899 Well that's the same thing as saying 3 to sixth power is 161 00:08:36,899 --> 00:08:40,110 equal to 729, because that's all the negative exponent 162 00:08:40,110 --> 00:08:41,759 does is inverts it. 163 00:08:41,759 --> 00:08:42,590 Let's see. 164 00:08:42,590 --> 00:08:44,879 We could multiply that out, but that should be the case. 165 00:08:44,879 --> 00:08:46,480 Because, well, we could look here. 166 00:08:46,480 --> 00:08:47,320 But let's see. 167 00:08:47,320 --> 00:08:52,629 3 to the third power-- This would be 3 to the third power 168 00:08:52,629 --> 00:08:57,450 times 3 to the third power is equal to 27 times 27. 169 00:08:57,450 --> 00:08:58,540 That looks pretty close. 170 00:08:58,539 --> 00:09:00,899 You can confirm it with a calculator if you 171 00:09:00,899 --> 00:09:02,340 don't believe me. 172 00:09:02,340 --> 00:09:04,940 Anyway, that's all the time I have in this video. 173 00:09:04,940 --> 00:09:07,500 In the next video, I'll introduce you to the last 174 00:09:07,500 --> 00:09:08,679 two logarithm properties. 175 00:09:08,679 --> 00:09:12,489 And, if we have time, maybe I'll do examples with 176 00:09:12,490 --> 00:09:13,440 the leftover time. 177 00:09:13,440 --> 00:09:15,120 I'll see you soon.