1 00:00:00,677 --> 00:00:03,067 I would guess that you are reasonably familiar 2 00:00:03,067 --> 00:00:04,313 with linear scales. 3 00:00:04,313 --> 00:00:07,182 These are the scales that you would typically see in most of your math classes, 4 00:00:07,533 --> 00:00:10,933 and so just to make sure we know what we are talking about 5 00:00:10,933 --> 00:00:12,800 and maybe thinking about in a slightly different way 6 00:00:12,800 --> 00:00:14,467 let me draw a linear number line. 7 00:00:14,467 --> 00:00:16,200 Let me start with the zero 8 00:00:16,200 --> 00:00:17,467 and what we are going to do is we are gonna say: 9 00:00:17,467 --> 00:00:19,467 look if I move this distance right over here 10 00:00:19,467 --> 00:00:23,867 and if I move that distance to the right, that's equivalent to adding 10 11 00:00:23,867 --> 00:00:27,800 so if you start at zero and you add 10, that will obviously get you a 10 12 00:00:27,800 --> 00:00:31,400 If If you move that distance to the right again, you're gonna add 10 again 13 00:00:31,400 --> 00:00:34,600 that will get you to 20 14 00:00:34,600 --> 00:00:37,400 and obviously we could keep doing it and get to 30, 40, 50 so on and so forth 15 00:00:37,400 --> 00:00:40,200 and also just looking at what we did here 16 00:00:40,200 --> 00:00:41,215 if we go the other direction 17 00:00:41,215 --> 00:00:44,662 if we start here and move that same distance to the left 18 00:00:44,662 --> 00:00:46,200 we're clearly subtracting 10 19 00:00:46,200 --> 00:00:49,200 10 minus 10 is equal to zero 20 00:00:49,200 --> 00:00:51,800 so if we move that distance to the left again, 21 00:00:51,800 --> 00:00:55,800 we would get to negative 10 and if we did it again we would get to negative 20 22 00:00:55,800 --> 00:01:00,200 so the general idea is, however many times we move that distance 23 00:01:00,200 --> 00:01:04,400 we are essentially adding or how many times we move that distance to the right 24 00:01:04,400 --> 00:01:06,667 we are essentially adding that multiple of ten 25 00:01:06,667 --> 00:01:08,867 if we move to it twice, we're adding 2 times 10 26 00:01:08,867 --> 00:01:13,000 and that not only works for whole numbers, it would work for fractions as well 27 00:01:13,000 --> 00:01:14,600 where would 5 be? 28 00:01:14,600 --> 00:01:18,400 Well, to get to 5, we only have to multiply 10 29 00:01:18,400 --> 00:01:21,000 or I guess one way to think about it is 10 or rather 30 00:01:21,000 --> 00:01:23,733 one way to think about it is 5 is half of 10 31 00:01:23,733 --> 00:01:25,800 and so if we want to only go half of ten 32 00:01:25,800 --> 00:01:27,400 we only have to go half this distance 33 00:01:27,400 --> 00:01:30,800 so if we go half this distance, 34 00:01:30,800 --> 00:01:32,800 if we go half this distance 35 00:01:32,800 --> 00:01:34,600 that will get us to one half times ten 36 00:01:34,600 --> 00:01:36,467 In this case that would be five 37 00:01:36,467 --> 00:01:37,667 If we go the left 38 00:01:37,667 --> 00:01:39,267 that would get us to negative five 39 00:01:39,267 --> 00:01:40,667 and there's nothing, 40 00:01:40,667 --> 00:01:43,200 let me draw that a little bit more centered, negative five 41 00:01:43,200 --> 00:01:44,800 and there's nothing really new here 42 00:01:44,800 --> 00:01:48,200 we're just kinda thinking about it in a slightly novel way that's going to be useful 43 00:01:48,200 --> 00:01:49,600 when we start thinking about logarithm 44 00:01:49,600 --> 00:01:51,800 but this is just the number line that you've always known 45 00:01:51,800 --> 00:01:54,200 if we want to put one here, we'd move one tenth of the distance 46 00:01:54,200 --> 00:01:55,600 cause one is one tenth of ten 47 00:01:55,600 --> 00:01:58,267 So this would be 1,2,3,4 48 00:01:58,267 --> 00:01:59,067 I could just put 49 00:01:59,067 --> 00:01:59,800 I could 50 00:01:59,800 --> 00:02:02,067 I could label frankly any, any number right over here 51 00:02:02,067 --> 00:02:04,667 Now this was the situation when we add 10 or subtract 10 52 00:02:04,667 --> 00:02:10,800 but it's completely legitimate to have an alternate way of thinking of what you do when you move this distance 53 00:02:10,800 --> 00:02:11,885 and let's think about that 54 00:02:12,333 --> 00:02:14,467 so let's say i have another line over here 55 00:02:14,467 --> 00:02:17,000 and you might guess this is going to be the logarithmic number line 56 00:02:17,000 --> 00:02:19,800 we give ourselves some space 57 00:02:19,800 --> 00:02:22,600 and let's start this logarithmic number line at 1 58 00:02:22,600 --> 00:02:26,400 and I'll let you think about, after this video, why I didn't start it at zero 59 00:02:26,400 --> 00:02:28,333 and if you start at 1 60 00:02:28,333 --> 00:02:29,733 and instead of moving that 61 00:02:29,733 --> 00:02:32,133 so I'm still going to define that same distance 62 00:02:32,133 --> 00:02:33,333 that same distance 63 00:02:33,333 --> 00:02:35,000 it's gonna be a little smaller 64 00:02:35,000 --> 00:02:37,000 i'm still gonna to define that same distance 65 00:02:37,000 --> 00:02:39,200 but instead of saying that that same distance is adding 10 66 00:02:39,200 --> 00:02:40,200 when I move to the right 67 00:02:40,200 --> 00:02:41,933 I'm gonna say when I move to the right 68 00:02:41,933 --> 00:02:44,600 that distance when I move to the right on this new number line that I've created 69 00:02:44,600 --> 00:02:47,733 that is the same thing as multiplying by 10 70 00:02:47,733 --> 00:02:49,533 so if I move that distance 71 00:02:49,533 --> 00:02:50,333 I start at 1 72 00:02:50,333 --> 00:02:51,333 I multiply by 10 73 00:02:51,333 --> 00:02:52,333 that gets me to 74 00:02:52,333 --> 00:02:54,333 that gets me to 10! 75 00:02:54,333 --> 00:02:56,133 and then if I multiply by 10 again 76 00:02:56,133 --> 00:02:57,867 if i multiply by 10 again 77 00:02:57,867 --> 00:02:59,013 If I move by that distance again, 78 00:02:59,013 --> 00:03:00,282 I'm multiplying by 10 again 79 00:03:00,282 --> 00:03:01,921 and so that would get me to 100 80 00:03:01,921 --> 00:03:04,729 and I think you can already see the difference that's happening 81 00:03:04,729 --> 00:03:07,175 and what about moving to the left that distance? 82 00:03:07,600 --> 00:03:09,800 Well we already kind have said what happens 83 00:03:09,800 --> 00:03:11,200 cause if start here 84 00:03:11,200 --> 00:03:12,331 we start at a 100 85 00:03:12,331 --> 00:03:13,431 and move to the left by that distance 86 00:03:13,431 --> 00:03:14,759 What happens? 87 00:03:14,759 --> 00:03:17,000 Well, I divided by 10 88 00:03:17,005 --> 00:03:19,708 100 divide by 10 gets me 10 89 00:03:19,708 --> 00:03:21,069 10 divided by 10 gives me 1 90 00:03:21,069 --> 00:03:24,933 and so if I move that distance to the left again 91 00:03:24,933 --> 00:03:27,000 I'll divide by 10 again 92 00:03:27,000 --> 00:03:28,656 that will get me to 93 00:03:28,656 --> 00:03:30,333 one tenth 94 00:03:30,333 --> 00:03:32,133 and if I move that distance to the left again 95 00:03:32,133 --> 00:03:38,400 that will get me to one over a hundred 96 00:03:38,400 --> 00:03:39,933 and so the general idea is 97 00:03:39,933 --> 00:03:42,800 is however many times I move that distance to the right 98 00:03:42,800 --> 00:03:48,200 I'm multiplying my starting point by 10 that many times 99 00:03:48,200 --> 00:03:50,538 and so for example 100 00:03:50,538 --> 00:03:52,867 when I move that distance twice 101 00:03:52,867 --> 00:03:55,333 so this whole distance right over here 102 00:03:55,333 --> 00:03:57,067 I went that distance twice 103 00:03:57,067 --> 00:03:59,733 so this is times 10 times 10 104 00:03:59,733 --> 00:04:03,867 which is the same thing as times 10 to the second power 105 00:04:03,867 --> 00:04:06,800 and so really, i'm raising 10 106 00:04:06,800 --> 00:04:12,267 I'm multiplying it, times 10 to whatever power however many times i'm jumping to the right 107 00:04:12,267 --> 00:04:13,267 Same thing 108 00:04:13,267 --> 00:04:17,944 If I go to the left that distance twice 109 00:04:17,944 --> 00:04:20,133 Let me do that in a new colour 110 00:04:20,133 --> 00:04:22,933 If I go to the left that distance twice 111 00:04:22,933 --> 00:04:26,267 this will be the same thing as dividing by 10 twice 112 00:04:26,267 --> 00:04:28,467 dividing by 10, dividing by 10 113 00:04:28,467 --> 00:04:31,667 which is the same thing as multiplying by, 114 00:04:31,667 --> 00:04:32,933 well one way to think of it 115 00:04:32,933 --> 00:04:35,933 1/10² 116 00:04:35,933 --> 00:04:37,800 or dividing by 10² 117 00:04:37,800 --> 00:04:39,467 is another way of thinking about it 118 00:04:39,467 --> 00:04:41,067 and so that might make a little 119 00:04:41,067 --> 00:04:41,667 you know 120 00:04:41,667 --> 00:04:42,667 that might be, hopefully, a little bit intuitive 121 00:04:42,667 --> 00:04:44,667 and you can already see why this is valuable 122 00:04:44,667 --> 00:04:46,667 we can already, on this number line, 123 00:04:46,667 --> 00:04:49,267 plot a much broader spectrum of things 124 00:04:49,267 --> 00:04:50,800 than we can on this number line 125 00:04:50,800 --> 00:04:52,067 we can go all the way up to a 100 126 00:04:52,067 --> 00:04:54,067 and then we even get some nice granularity 127 00:04:54,067 --> 00:04:56,000 if we want to go down to one tenth and one hundredth 128 00:04:56,000 --> 00:05:00,131 here we don't get the granularity at small scales 129 00:05:00,400 --> 00:05:02,333 and we also don't get to go to really large numbers 130 00:05:02,333 --> 00:05:04,200 and if we go a little distance more 131 00:05:04,200 --> 00:05:07,200 we get to 1000 and then we get to 10000 so on and so forth 132 00:05:07,200 --> 00:05:11,400 so we can really cover a much broader spectrum on this line right over here 133 00:05:11,400 --> 00:05:13,000 but what's also neat about this 134 00:05:13,000 --> 00:05:15,000 is that when you move a fixed distance 135 00:05:15,000 --> 00:05:17,867 so when you move a fixed distance on this linear number line 136 00:05:17,867 --> 00:05:20,133 you are adding or subtracting that amount 137 00:05:20,133 --> 00:05:23,533 so if you move that fixed distance you are adding to, to the right 138 00:05:23,533 --> 00:05:25,333 if yo go to the left you're subtracting to 139 00:05:25,333 --> 00:05:28,333 when you do the same thing on a logarithmic number line 140 00:05:28,333 --> 00:05:29,867 and this is true of any logarithmic number line 141 00:05:29,867 --> 00:05:32,333 you will be scaling by a fixed factor 142 00:05:32,333 --> 00:05:34,933 and one way to think about what that fixed factor is 143 00:05:34,933 --> 00:05:37,800 is this idea of exponents 144 00:05:37,800 --> 00:05:39,800 so if you wanted to say 145 00:05:39,800 --> 00:05:42,800 Where would 2 sit on this number line? 146 00:05:42,800 --> 00:05:45,400 Then you would just think to yourself 147 00:05:45,400 --> 00:05:48,800 well if i asked myself, where does 100 sit on that number line? 148 00:05:48,800 --> 00:05:50,400 and actually, that might be a better place to start 149 00:05:50,400 --> 00:05:52,000 if i said, If I hadn't already plot it 150 00:05:52,000 --> 00:05:53,333 Where does 100 sit on that number line? 151 00:05:53,333 --> 00:05:57,533 I'd say, how many times do I have to multiply 10 by itself to get 100? 152 00:05:57,533 --> 00:06:00,400 and that's how many times I need to move this distance 153 00:06:00,400 --> 00:06:01,933 and so essentially I would be asking 154 00:06:01,933 --> 00:06:05,800 10 to the what power is equal to 100 155 00:06:05,800 --> 00:06:08,200 and then I would get that 'question mark' is equal to 2 156 00:06:08,200 --> 00:06:11,200 and then I would move that many spaces to plot my 100 157 00:06:11,200 --> 00:06:13,667 another way of stating this exact same thing is 158 00:06:13,667 --> 00:06:18,000 log base 10 of 100 is equal to 'question mark' 159 00:06:18,000 --> 00:06:20,667 and this 'question mark' is clearly equal to 2 160 00:06:20,667 --> 00:06:26,000 and that says I need to plot 100 to 2 of this distance to the right 161 00:06:26,000 --> 00:06:29,595 and to figure out where would I plot the 2 I would do the same exact same thing 162 00:06:29,595 --> 00:06:30,318 I would say 163 00:06:30,318 --> 00:06:36,000 10 to what power is equal to 2? 164 00:06:36,000 --> 00:06:40,667 or log base 10 of 2 is equal to what? 165 00:06:40,667 --> 00:06:43,969 and we can get the trusty calculator out 166 00:06:43,969 --> 00:06:46,462 and we can just say log 167 00:06:46,462 --> 00:06:49,600 and on most calculators it's just a log without the base specified 168 00:06:49,600 --> 00:06:50,733 they're assuming base 10 169 00:06:50,733 --> 00:06:55,600 so log of 2 is equal to roughly 0.3 170 00:06:55,600 --> 00:06:57,200 0.301 171 00:06:57,250 --> 00:07:01,533 so this is equal to 0.301 172 00:07:01,533 --> 00:07:03,133 so what this tells us is 173 00:07:03,133 --> 00:07:06,733 we need to move this fraction of this distance to get to 2 174 00:07:06,733 --> 00:07:07,846 If we move this whole distance 175 00:07:07,846 --> 00:07:10,533 it's like multiplying 10 times 10 to the first power 176 00:07:10,533 --> 00:07:15,667 but since we only get 10 to the 0.301 power, we only want to do 0.301 of this distance 177 00:07:15,667 --> 00:07:17,667 so it's going to be roughly a third of this 178 00:07:17,667 --> 00:07:19,267 so let me 179 00:07:19,267 --> 00:07:20,267 it's going to be roughly 180 00:07:20,267 --> 00:07:22,133 actually a little less than a third 181 00:07:22,133 --> 00:07:23,333 0.3 not 0.33 182 00:07:23,333 --> 00:07:24,467 so 2 is going to sit 183 00:07:24,467 --> 00:07:25,800 2 is going to 184 00:07:25,800 --> 00:07:27,333 let me do it a little more to the right 185 00:07:27,333 --> 00:07:29,467 so 2 is going to sit right over here 186 00:07:29,467 --> 00:07:31,133 now what's really cool about it is 187 00:07:31,133 --> 00:07:33,800 this distance in general, on this logarithmic number line 188 00:07:33,800 --> 00:07:37,308 means multiplying by 2 189 00:07:37,308 --> 00:07:40,267 and so if you go that same distance again 190 00:07:40,267 --> 00:07:41,467 you're gonna get to 4 191 00:07:41,467 --> 00:07:45,267 if you multiply that same distance again, you're going to multiply by 4 192 00:07:45,267 --> 00:07:49,933 and if you go that same distance again, you are going to get to 8 193 00:07:49,933 --> 00:07:51,000 and so if you said well 194 00:07:51,000 --> 00:07:55,600 Where would I plot 5 on this number line? 195 00:07:55,600 --> 00:07:57,800 Well there's a couple of ways to do it. 196 00:07:57,800 --> 00:08:01,200 You could really figure out what the base 10 logarithm of 5 is 197 00:08:01,200 --> 00:08:03,200 and figure out where it goes on the number line 198 00:08:03,200 --> 00:08:04,733 or you could say look! 199 00:08:04,733 --> 00:08:08,595 If I start at 10 200 00:08:08,595 --> 00:08:10,533 and if I move this distance to the left 201 00:08:10,533 --> 00:08:12,533 I'm going to be dividing by 2 202 00:08:12,533 --> 00:08:16,467 so if I move this distance to the left I will be dividing by 2 203 00:08:16,467 --> 00:08:18,133 I know it's getting a little bit messy here 204 00:08:18,133 --> 00:08:20,867 i'll maybe do another video where we learn how to draw a clean version of this 205 00:08:20,867 --> 00:08:24,733 so if I start at 10 and then go that same distance I'm dividing by 2 206 00:08:24,733 --> 00:08:27,733 and so this right here would be 207 00:08:27,733 --> 00:08:30,333 that right over there would be 5 208 00:08:30,333 --> 00:08:31,733 Now the next question you say 209 00:08:31,733 --> 00:08:33,067 Where do I plot 3? 210 00:08:33,067 --> 00:08:34,733 Well we can do the exact same thing that we did with 2 211 00:08:34,733 --> 00:08:37,302 we ask ourselves 212 00:08:37,302 --> 00:08:41,569 what power do we have to raise 10 to, to get to 3 213 00:08:41,569 --> 00:08:42,800 and to get that 214 00:08:42,800 --> 00:08:44,200 we once again get our calculator out 215 00:08:44,200 --> 00:08:48,667 log base 10 of 3 is equal to 0.477 216 00:08:48,667 --> 00:08:50,667 so it's almost halfway 217 00:08:50,667 --> 00:08:53,267 so it's almost going to be half of this distance 218 00:08:53,267 --> 00:08:56,867 so half of that distance is gonna look something like right over there 219 00:08:56,867 --> 00:09:00,600 so 3 is going to go right over here 220 00:09:00,600 --> 00:09:02,267 and you could do the logarithm 221 00:09:02,267 --> 00:09:04,000 let's see we're missing 6, 7 and 8 222 00:09:04,000 --> 00:09:05,067 oh we have 8 223 00:09:05,067 --> 00:09:05,800 we're missing 9 224 00:09:05,800 --> 00:09:09,000 so then to get 9, we just have to mutiply by 3 again 225 00:09:09,000 --> 00:09:10,267 so this is 3 226 00:09:10,267 --> 00:09:12,000 and if we go that same distance 227 00:09:12,000 --> 00:09:13,400 we multiply by 3 again 228 00:09:13,400 --> 00:09:16,200 9 is gonna be squeezed in right over here 229 00:09:16,200 --> 00:09:18,267 9 is gonna be squeezed in right over there 230 00:09:18,267 --> 00:09:19,800 and if we wanna get to 6 231 00:09:19,800 --> 00:09:22,067 we just have to multiply by 2 232 00:09:22,067 --> 00:09:24,067 and we already know the distance to multiply by 2 233 00:09:24,067 --> 00:09:26,467 it's this thing right over here 234 00:09:26,467 --> 00:09:28,400 so you multiply that by 2 235 00:09:28,400 --> 00:09:32,200 you do that same distance and you're gonna get to 6 236 00:09:32,200 --> 00:09:34,667 and if you wanted to figure out where 7 is 237 00:09:34,667 --> 00:09:39,323 once again you could take the log 238 00:09:39,323 --> 00:09:40,792 let me do it right over here 239 00:09:40,792 --> 00:09:43,267 so you take the log of 7 240 00:09:43,267 --> 00:09:46,200 it is going to be roughly 0.85 241 00:09:46,200 --> 00:09:48,267 so 7 is just going to be squeezed in 242 00:09:48,267 --> 00:09:51,067 roughly right over there 243 00:09:51,067 --> 00:09:53,400 so a couple of neat things you already appreciated 244 00:09:53,400 --> 00:09:55,667 one, we can fit more on this logarithmic scale 245 00:09:55,667 --> 00:09:59,800 and, as i did with the video with Vi Hart 246 00:09:59,800 --> 00:10:02,667 where she talked about how we perceive many things with logarithmic scales 247 00:10:02,667 --> 00:10:06,800 so that is actually a good way to even understand some of human perception 248 00:10:06,800 --> 00:10:10,333 but the other really cool thing is when we move a fixed distance on this logarithmic scale 249 00:10:10,333 --> 00:10:13,200 we are multiplying by a fixed constant 250 00:10:13,200 --> 00:10:16,933 now the one kind of strange thing about this and you might have already noticed here 251 00:10:16,933 --> 00:10:20,600 is that we don't see the numbers lined up the way we normally see them 252 00:10:20,600 --> 00:10:22,333 there is a big jump from 1 to 2 253 00:10:22,333 --> 00:10:24,200 then a smaller jump from 3 to 4 254 00:10:24,200 --> 00:10:25,600 then a smaller jump from that from 3 to 4 255 00:10:25,600 --> 00:10:27,333 then even smaller from 4 to 5 256 00:10:27,333 --> 00:10:28,533 then even smaller from 5 to 6 257 00:10:28,533 --> 00:10:30,600 and then 7, 8, 9 258 00:10:30,600 --> 00:10:32,133 7 is gonna be right in there 259 00:10:32,133 --> 00:10:33,400 it gets squeezed squeezed squeezed in 260 00:10:33,400 --> 00:10:34,600 tighter and tighter and tighter 261 00:10:34,600 --> 00:10:35,533 and then you get 10 262 00:10:35,533 --> 00:10:36,933 and then you get another big jump 263 00:10:36,933 --> 00:10:40,600 because once again if you wanna get to 20, you just have to multiply by 2 264 00:10:40,600 --> 00:10:43,333 you just have to multiply by 2 again 265 00:10:43,333 --> 00:10:45,800 so this distance again gets us to 20 266 00:10:45,800 --> 00:10:50,800 if you go this distance over here that will get you to 30 267 00:10:50,800 --> 00:10:52,667 cause you're multiplying by 3 268 00:10:52,667 --> 00:10:54,800 so this right over here is our times 3 distance 269 00:10:54,800 --> 00:10:58,600 so if you do that again, if you do that distance 270 00:10:58,600 --> 00:10:59,800 then that gets you to 30 271 00:10:59,800 --> 00:11:01,000 you're multiplying by 3 272 00:11:01,000 --> 00:11:03,933 and then you could plot the whole same thing over here again 273 00:11:03,933 --> 00:11:07,400 but hopefully this gives you a little bit more intuition of why logarithmic number lines 274 00:11:07,400 --> 00:11:07,908 look the way they do 275 00:11:07,908 --> 00:11:10,200 or why logarithmic scale looks the way it does 276 00:11:10,200 --> 99:59:59,999 and also it gives you a little bit of appreciation for why they might be useful.