1 00:00:07,600 --> 00:00:07,860 PROFESSOR: OK. 2 00:00:07,860 --> 00:00:08,350 Hi. 3 00:00:08,350 --> 00:00:13,620 I thought I'd give a short lecture about how logarithms 4 00:00:13,620 --> 00:00:15,230 are actually used. 5 00:00:15,230 --> 00:00:17,760 So a little bit practical. 6 00:00:17,760 --> 00:00:22,750 And also, it naturally comes in, how quickly 7 00:00:22,750 --> 00:00:24,720 do functions grow? 8 00:00:24,720 --> 00:00:28,350 Which functions grow faster than others? 9 00:00:28,350 --> 00:00:33,700 And I made a list of a bunch of functions that 10 00:00:33,700 --> 00:00:35,290 we see all the time. 11 00:00:35,290 --> 00:00:37,520 Linear growth. 12 00:00:37,520 --> 00:00:41,440 Just, the function goes up along the straight line. 13 00:00:41,440 --> 00:00:46,230 Proportional to x, linear could have been a c times x, 14 00:00:46,230 --> 00:00:48,240 still linear. 15 00:00:48,240 --> 00:00:54,990 Here that's called polynomial growth, like some power of x. 16 00:00:54,990 --> 00:00:58,080 Here is faster growth. 17 00:00:58,080 --> 00:01:02,100 We introduced e to the x, and I'll take this chance to bring 18 00:01:02,100 --> 00:01:05,300 in 2 to the x and 10 to the x. 19 00:01:05,300 --> 00:01:08,180 Especially 10 to the x, because that'll lead us to 20 00:01:08,180 --> 00:01:16,110 logarithms to base 10, and those are handy in practice. 21 00:01:16,110 --> 00:01:18,040 So that's exponential growth. 22 00:01:18,040 --> 00:01:21,680 And here are some that grow faster still. 23 00:01:21,680 --> 00:01:27,655 x factorial, n factorial grows really fast. And n to the nth 24 00:01:27,655 --> 00:01:31,810 or x to the xth is a function that grows still faster. 25 00:01:31,810 --> 00:01:34,960 And of course, we could cook up a function the 26 00:01:34,960 --> 00:01:36,140 grew faster than that. 27 00:01:36,140 --> 00:01:40,680 X to the x to the x power would really just take off. 28 00:01:40,680 --> 00:01:44,180 And we could find functions that grow more slowly. 29 00:01:44,180 --> 00:01:50,770 But let's just take these and let x be 1000. 30 00:01:50,770 --> 00:01:54,770 Just to have a kind of realistic idea of how these 31 00:01:54,770 --> 00:01:57,320 compare when x is 1000. 32 00:01:57,320 --> 00:01:57,870 OK. 33 00:01:57,870 --> 00:02:00,060 So I'm skipping to c. 34 00:02:00,060 --> 00:02:02,390 So x will be 1000. 35 00:02:02,390 --> 00:02:04,360 10 cubed. 36 00:02:04,360 --> 00:02:11,810 Let me just write it as 10 cubed. 37 00:02:11,810 --> 00:02:15,940 So x is going to be 1000. 38 00:02:15,940 --> 00:02:18,430 And because these are big numbers, I'm going to write 39 00:02:18,430 --> 00:02:21,170 them as powers of 10. 40 00:02:21,170 --> 00:02:22,280 OK. 41 00:02:22,280 --> 00:02:25,460 so how about 1000 squared? 42 00:02:25,460 --> 00:02:34,560 10 cubed squared will be 10 to the sixth. 43 00:02:34,560 --> 00:02:40,030 1000 cubed, we're up to 10 to the ninth. 44 00:02:40,030 --> 00:02:41,500 And onwards. 45 00:02:41,500 --> 00:02:45,420 Like, this is where the economists are working. 46 00:02:45,420 --> 00:02:49,460 The national debt is in this range. 47 00:02:49,460 --> 00:02:49,840 OK. 48 00:02:49,840 --> 00:02:54,070 Now fortunately, it's not in this range. 49 00:02:54,070 --> 00:02:56,640 2 to the thousandth power. 50 00:02:56,640 --> 00:02:59,620 And if I want to be able to compare it, I'll write that 51 00:02:59,620 --> 00:03:03,540 approximately as 10 to-- 52 00:03:03,540 --> 00:03:07,710 well, if it's 2 to the thousandth power, it'll be 10 53 00:03:07,710 --> 00:03:09,850 to a smaller power. 54 00:03:09,850 --> 00:03:16,980 And 300 is pretty close for 2 to the thousandth. 55 00:03:16,980 --> 00:03:21,150 Then e to the thousandth, that's going to be bigger than 56 00:03:21,150 --> 00:03:24,890 2. e is 2.7 et cetera. 57 00:03:24,890 --> 00:03:27,300 This is more like 10 to the-- 58 00:03:27,300 --> 00:03:33,770 I think this is right-- about 434, maybe. 59 00:03:33,770 --> 00:03:36,410 And 10 to the thousandth-- 60 00:03:36,410 --> 00:03:37,960 well, I can write that right in. 61 00:03:37,960 --> 00:03:42,380 10 to the thousandth when x is 1000. 62 00:03:42,380 --> 00:03:43,370 OK. 63 00:03:43,370 --> 00:03:46,000 So that's the one that is exactly right. 64 00:03:46,000 --> 00:03:51,540 And also, I could write in 1000 to the thousandth power. 65 00:03:51,540 --> 00:03:54,260 What power of 10 will this be? 66 00:03:54,260 --> 00:03:56,470 10 to the what? 67 00:03:56,470 --> 00:04:00,730 1000 to the thousandth power, I think, is 10 to the three 68 00:04:00,730 --> 00:04:02,690 thousandth. 69 00:04:02,690 --> 00:04:04,850 Why do I think that? 70 00:04:04,850 --> 00:04:10,920 Because 1000 itself is 10 times 10 times 10. 71 00:04:10,920 --> 00:04:13,180 Three of them, right? 72 00:04:13,180 --> 00:04:17,779 And then we do that 1000 times, so we have a string of 73 00:04:17,779 --> 00:04:21,620 3000 10s multiplying each other. 74 00:04:21,620 --> 00:04:23,960 And that's what 10 to the three thousandth is. 75 00:04:23,960 --> 00:04:29,120 And you might wonder about a thousand factorial. 76 00:04:29,120 --> 00:04:31,695 Let me make the rough estimate. 77 00:04:34,220 --> 00:04:38,010 A big number in factorial, order of magnitude, is 78 00:04:38,010 --> 00:04:41,220 something like, it doesn't grow as fast as this, because 79 00:04:41,220 --> 00:04:45,480 this is x times x minus 1 times x minus 2. 80 00:04:45,480 --> 00:04:48,990 1000 times 999 times 998. 81 00:04:48,990 --> 00:04:52,710 So we're not repeating 1000 every time. 82 00:04:52,710 --> 00:04:54,150 And the difference-- 83 00:04:54,150 --> 00:04:59,630 it turns out that this number divided by this number, x to 84 00:04:59,630 --> 00:05:03,130 the x over e to the x, is the right 85 00:05:03,130 --> 00:05:05,700 general picture for factorial. 86 00:05:05,700 --> 00:05:10,490 So that would be, if I divide 10 to the 3000 by 10 and this 87 00:05:10,490 --> 00:05:11,550 power, what do I do? 88 00:05:11,550 --> 00:05:14,320 In a division, I do a subtraction of exponents, 89 00:05:14,320 --> 00:05:18,530 because I have that many fewer 10s multiplying each other. 90 00:05:18,530 --> 00:05:22,300 So I think it would be 3000, but I don't want the full 91 00:05:22,300 --> 00:05:27,530 3000, because I take away e to the thousandth, 434 of them. 92 00:05:27,530 --> 00:05:28,520 So that's about-- 93 00:05:28,520 --> 00:05:34,750 2566 is close enough, anyway. 94 00:05:34,750 --> 00:05:35,190 OK. 95 00:05:35,190 --> 00:05:36,440 Giant numbers. 96 00:05:36,440 --> 00:05:37,320 Giant numbers. 97 00:05:37,320 --> 00:05:41,970 And of course you saw that I didn't write it out with 1 and 98 00:05:41,970 --> 00:05:43,375 3000, or whatever, zeros. 99 00:05:43,375 --> 00:05:43,530 Hopeless. 100 00:05:43,530 --> 00:05:44,780 OK. 101 00:05:46,760 --> 00:05:51,800 In other words, it's the exponent that gives me 102 00:05:51,800 --> 00:05:53,350 something I can really work with. 103 00:05:53,350 --> 00:05:55,850 And the exponent is the logarithm. 104 00:05:55,850 --> 00:05:57,150 That's what logarithms are. 105 00:05:57,150 --> 00:05:58,690 They are the exponents. 106 00:05:58,690 --> 00:06:05,190 And when they're the exponent with a 10, I call 10 the base. 107 00:06:05,190 --> 00:06:09,290 And I'm speaking about logarithms to the base 10. 108 00:06:09,290 --> 00:06:11,850 Can I just copy those numbers again? 109 00:06:11,850 --> 00:06:14,250 And then I want to write their logarithms. Because it's the 110 00:06:14,250 --> 00:06:21,210 logarithms that kind of remain reasonable-looking numbers but 111 00:06:21,210 --> 00:06:25,180 tell you very nicely what's growing fast. 112 00:06:25,180 --> 00:06:27,340 So let me write out again. 113 00:06:27,340 --> 00:06:33,380 10 cubed, 10 sixth, 10 to the ninth is polynomial growth 114 00:06:33,380 --> 00:06:35,670 starting with the first power. 115 00:06:35,670 --> 00:06:38,800 Then I'll write down 10 to the three hundredth, 116 00:06:38,800 --> 00:06:40,280 approximately. 117 00:06:40,280 --> 00:06:43,920 10 to the 434, I think, is about right. 118 00:06:43,920 --> 00:06:46,760 And then 10 to the 1000. 119 00:06:46,760 --> 00:06:53,910 And then I had 10 to the 2566 as something, roughly 1000 120 00:06:53,910 --> 00:06:59,370 factorial, and then 10 to the 3000. 121 00:06:59,370 --> 00:06:59,660 OK. 122 00:06:59,660 --> 00:07:02,120 I just copied those numbers again. 123 00:07:02,120 --> 00:07:06,660 And now I plan to take their logarithms. I can see what's 124 00:07:06,660 --> 00:07:10,360 happening with logarithms. The logarithm of 10 125 00:07:10,360 --> 00:07:12,750 to the ninth is-- 126 00:07:12,750 --> 00:07:14,090 if the base is 10-- 127 00:07:14,090 --> 00:07:19,070 the logarithm of 10 to the ninth is the nine. 128 00:07:19,070 --> 00:07:21,170 This has logarithm 6. 129 00:07:21,170 --> 00:07:23,000 This has logarithm 3. 130 00:07:23,000 --> 00:07:24,700 So you see-- 131 00:07:24,700 --> 00:07:24,970 well. 132 00:07:24,970 --> 00:07:27,220 If we took the logarithm of the national debt, it wouldn't 133 00:07:27,220 --> 00:07:29,250 look too serious. 134 00:07:29,250 --> 00:07:34,540 It would just be up around 9 moving toward 10. 135 00:07:34,540 --> 00:07:39,510 But what I'm using it for here is to get some 136 00:07:39,510 --> 00:07:41,990 reasonable way to see-- 137 00:07:41,990 --> 00:07:42,840 300. 138 00:07:42,840 --> 00:07:44,355 Of course, that's big. 139 00:07:44,355 --> 00:07:46,370 For a logarithm, that's a very big number. 140 00:07:46,370 --> 00:07:49,190 434, 1000. 141 00:07:49,190 --> 00:07:51,310 These are climbing up. 142 00:07:51,310 --> 00:07:55,910 2566 and 3000. 143 00:07:55,910 --> 00:07:58,280 OK. 144 00:07:58,280 --> 00:07:59,530 So these are the logs. 145 00:08:02,270 --> 00:08:04,100 Just to repeat. 146 00:08:04,100 --> 00:08:08,200 If I wanted this growth, this list of functions by how fast 147 00:08:08,200 --> 00:08:11,440 they grow, where would log x appear 148 00:08:11,440 --> 00:08:13,580 in my list of functions? 149 00:08:13,580 --> 00:08:16,310 It would be way at the left end. 150 00:08:16,310 --> 00:08:17,840 Slower than x. 151 00:08:17,840 --> 00:08:18,980 Much slower than x. 152 00:08:18,980 --> 00:08:23,040 Log x grows very slowly, as we see here. 153 00:08:23,040 --> 00:08:25,590 And then if you wanted one that really grew slowly, it 154 00:08:25,590 --> 00:08:28,080 would be log of log x. 155 00:08:28,080 --> 00:08:30,180 That creeps along. 156 00:08:30,180 --> 00:08:32,590 Eventually gets to-- 157 00:08:32,590 --> 00:08:33,940 passes any number. 158 00:08:33,940 --> 00:08:36,400 But x has to be enormous. 159 00:08:36,400 --> 00:08:40,946 And one more little comment before I begin to use some 160 00:08:40,946 --> 00:08:43,120 things graphically. 161 00:08:43,120 --> 00:08:48,445 Because that's the other part of this talk, is log-- 162 00:08:48,445 --> 00:08:49,850 the graphs. 163 00:08:49,850 --> 00:08:51,810 Using logarithms in graphs. 164 00:08:51,810 --> 00:08:53,240 A little point. 165 00:08:53,240 --> 00:08:57,340 You might ask, what about functions that decay? 166 00:08:57,340 --> 00:09:00,850 What would be the corresponding functions here 167 00:09:00,850 --> 00:09:01,890 that decay? 168 00:09:01,890 --> 00:09:03,460 Let me write them here. 169 00:09:03,460 --> 00:09:04,710 Decay. 170 00:09:06,700 --> 00:09:09,610 By that I mean, headed for 0 instead 171 00:09:09,610 --> 00:09:11,110 of headed for infinity. 172 00:09:11,110 --> 00:09:17,110 Well, 1 over x, 1 over x squared, 1 over x cubed. 173 00:09:20,300 --> 00:09:27,080 Those functions go to 0 faster and faster. 174 00:09:27,080 --> 00:09:29,410 Now, what about these? 175 00:09:29,410 --> 00:09:33,190 The next list would be 1 over-- 176 00:09:33,190 --> 00:09:36,070 I'm dividing, but 1 over 2 to the x. 177 00:09:36,070 --> 00:09:37,370 1 over e to the x. 178 00:09:37,370 --> 00:09:38,970 Can I write that in a better way? 179 00:09:38,970 --> 00:09:40,880 e to the minus x. 180 00:09:40,880 --> 00:09:45,980 1 over 10 to the x. 181 00:09:45,980 --> 00:09:49,290 Those are going to 0 like crazy. 182 00:09:49,290 --> 00:09:52,730 And of course, if I keep going, even worse. 183 00:09:52,730 --> 00:09:59,430 So like, x to the minus x power would be really small. 184 00:09:59,430 --> 00:10:05,220 So my point is just that we have a scale here that not 185 00:10:05,220 --> 00:10:09,720 only gives us a handle of how to deal with things that are 186 00:10:09,720 --> 00:10:13,130 growing very fast, but also things that are going to 0 187 00:10:13,130 --> 00:10:16,920 very fast. The other, the negative logarithms. The 188 00:10:16,920 --> 00:10:20,590 logarithms of these things would be minus 3, minus 6, 189 00:10:20,590 --> 00:10:25,260 minus 9 and so on, if I divide by one. 190 00:10:25,260 --> 00:10:26,120 Good. 191 00:10:26,120 --> 00:10:26,630 All right. 192 00:10:26,630 --> 00:10:29,610 So that suggests the idea. 193 00:10:29,610 --> 00:10:36,000 Now I want to introduce the idea of a log scale. 194 00:10:36,000 --> 00:10:48,475 So I'm just going to think of a usual straight line, on 195 00:10:48,475 --> 00:10:54,290 which we usually mark out 0, 1, 2, 3, minus 1, minus 2. 196 00:10:54,290 --> 00:11:04,170 But on this log scale, the center point, the 0, I'm 197 00:11:04,170 --> 00:11:08,140 really graphing the logarithm of x instead of x. 198 00:11:08,140 --> 00:11:09,250 That's the point. 199 00:11:09,250 --> 00:11:12,640 That in this log scale, what I'm picturing 200 00:11:12,640 --> 00:11:13,890 along here will be-- 201 00:11:16,210 --> 00:11:22,680 this number will be 10 to the 0 power, which is 1. 202 00:11:22,680 --> 00:11:25,270 The next one will be 10. 203 00:11:25,270 --> 00:11:27,180 The next one will be 100. 204 00:11:27,180 --> 00:11:29,890 The next one will be 1000. 205 00:11:29,890 --> 00:11:34,850 So you see, within this picture-- 206 00:11:34,850 --> 00:11:37,440 on a graph that we could draw and look at 207 00:11:37,440 --> 00:11:39,370 on a printed page-- 208 00:11:39,370 --> 00:11:44,000 we can get big numbers by going from the ordinary 1, 2, 209 00:11:44,000 --> 00:11:48,900 3 scale to the log scale, which puts these points in 210 00:11:48,900 --> 00:11:49,430 this order. 211 00:11:49,430 --> 00:11:51,150 And let me put some of the other ones. 212 00:11:51,150 --> 00:11:54,080 Now, what one point goes there? 213 00:11:54,080 --> 00:11:55,860 1/10. 214 00:11:55,860 --> 00:12:00,640 Every time I go that far, I'm multiplying by 10. 215 00:12:00,640 --> 00:12:02,850 When I go this way, I'm dividing by 10. 216 00:12:02,850 --> 00:12:07,970 Up there, this is the number 1/10, which is the same as 10 217 00:12:07,970 --> 00:12:10,720 to the minus 1 power, right? 218 00:12:10,720 --> 00:12:14,850 Here is one hundredth. 219 00:12:14,850 --> 00:12:18,080 Here is one thousandth. 220 00:12:18,080 --> 00:12:18,610 And so on. 221 00:12:18,610 --> 00:12:24,820 So this log scale is able to deal with very small numbers 222 00:12:24,820 --> 00:12:30,540 and very large numbers in a reasonable way. 223 00:12:30,540 --> 00:12:35,530 And everybody sees the point here that really, what it is 224 00:12:35,530 --> 00:12:39,170 is the logarithms. So this is 0. 225 00:12:39,170 --> 00:12:43,550 This is 1, 2, 3, and so on. 226 00:12:43,550 --> 00:12:46,500 Minus 1, minus 2, minus 3. 227 00:12:46,500 --> 00:12:51,350 If I'm graphing, really, these are the logarithms of x. 228 00:12:51,350 --> 00:12:55,470 And I'm doing logs to base 10 again, because that gives us 229 00:12:55,470 --> 00:12:57,630 nice numbers. 230 00:12:57,630 --> 00:12:57,705 OK. 231 00:12:57,705 --> 00:13:00,560 By the way, what's that number? 232 00:13:00,560 --> 00:13:04,355 What's that number, halfway between there and there? 233 00:13:08,070 --> 00:13:13,910 It's not halfway between 1 and 10 in the ordinary sense, 234 00:13:13,910 --> 00:13:15,760 which is whatever, 5 and a half. 235 00:13:15,760 --> 00:13:17,020 No way. 236 00:13:17,020 --> 00:13:20,520 Halfway between here is-- 237 00:13:20,520 --> 00:13:21,580 you know what it will be? 238 00:13:21,580 --> 00:13:25,870 It'll be square root of 10. 239 00:13:25,870 --> 00:13:27,840 10 to the 1/2 power. 240 00:13:27,840 --> 00:13:33,720 The half is here. 241 00:13:33,720 --> 00:13:36,720 The log is a half, so the number is the 242 00:13:36,720 --> 00:13:37,490 square root of 10. 243 00:13:37,490 --> 00:13:40,760 That's about 3, a little more than 3. 244 00:13:40,760 --> 00:13:45,000 And what would be here, would be 10 to the minus 1/2. 245 00:13:45,000 --> 00:13:47,450 1 over square root of 10. 246 00:13:47,450 --> 00:13:49,800 So you see that picture. 247 00:13:49,800 --> 00:13:53,840 Oh, I have another question, before I use the scales. 248 00:13:53,840 --> 00:13:56,900 What if I like the powers of 2 better? 249 00:13:56,900 --> 00:14:00,320 In many cases, we might prefer powers of 2. 250 00:14:00,320 --> 00:14:04,370 Well, if I plotted the numbers-- 251 00:14:04,370 --> 00:14:07,620 I'm looking at this log scale. 252 00:14:07,620 --> 00:14:13,050 And suppose I plot the numbers 1, 2, 4, 8, whatever. 253 00:14:13,050 --> 00:14:14,720 16. 254 00:14:14,720 --> 00:14:16,520 What could you tell me about those? 255 00:14:16,520 --> 00:14:17,890 Well, I know where 1 is. 256 00:14:17,890 --> 00:14:19,370 It's right there. 257 00:14:19,370 --> 00:14:20,780 That's a 1. 258 00:14:20,780 --> 00:14:22,900 Well, two would be a little further over. 259 00:14:22,900 --> 00:14:26,590 Then 4, then 8 would come before 10, and 16 260 00:14:26,590 --> 00:14:28,280 would come after 10. 261 00:14:28,280 --> 00:14:32,020 I pointed there, but 16 would not come there. 262 00:14:32,020 --> 00:14:37,390 16 would be a lot closer, I think, in here. 263 00:14:37,390 --> 00:14:43,720 What's the deal with 1, 2, 4, 8, 16 on this log scale? 264 00:14:43,720 --> 00:14:45,970 They would be equally spaced. 265 00:14:48,790 --> 00:14:50,350 Of course, the spacing would be 266 00:14:50,350 --> 00:14:52,810 smaller than the 10 spacing. 267 00:14:52,810 --> 00:14:58,520 If every time I multiplied by 2, I go the same distance. 268 00:14:58,520 --> 00:15:01,500 After I'd done it about 10 times-- 269 00:15:01,500 --> 00:15:05,690 multiplied by 2 10 times-- so that's 2 to the tenth power is 270 00:15:05,690 --> 00:15:07,060 close to 1000. 271 00:15:07,060 --> 00:15:11,445 So 10 powers of 2 would bring me pretty near there. 272 00:15:11,445 --> 00:15:13,760 Anyway. 273 00:15:13,760 --> 00:15:15,830 And here's one more question. 274 00:15:15,830 --> 00:15:17,850 Where is 0? 275 00:15:17,850 --> 00:15:25,140 If my value that I wanted to plot happened to be 0, where 276 00:15:25,140 --> 00:15:28,530 is it on this graph? 277 00:15:28,530 --> 00:15:30,670 It's not there. 278 00:15:30,670 --> 00:15:34,270 You can't plot 0 on a log scale. 279 00:15:34,270 --> 00:15:37,780 It's way down at the-- 280 00:15:37,780 --> 00:15:41,620 you know, it's at the minus infinity end of the graph. 281 00:15:41,620 --> 00:15:48,390 Infinity is up there at that end, and 0 is down here. 282 00:15:48,390 --> 00:15:49,800 OK. 283 00:15:49,800 --> 00:15:50,330 Good. 284 00:15:50,330 --> 00:15:51,930 So can we use that log scale? 285 00:15:51,930 --> 00:15:53,780 How do we use that log scale? 286 00:15:53,780 --> 00:15:57,790 Let me give you an idea for what use that 287 00:15:57,790 --> 00:15:59,710 log scale might be. 288 00:15:59,710 --> 00:16:01,520 Practical use. 289 00:16:01,520 --> 00:16:08,600 Suppose I know, or have reason to believe, that my function 290 00:16:08,600 --> 00:16:17,580 might be of the form y is something times x to the nth. 291 00:16:17,580 --> 00:16:22,030 I have some quantity y. 292 00:16:22,030 --> 00:16:25,880 The output when the input is x. 293 00:16:25,880 --> 00:16:31,450 But I don't know these, that number a. 294 00:16:31,450 --> 00:16:32,960 So I've done an experiment. 295 00:16:32,960 --> 00:16:35,930 And I would like to know what is a, and 296 00:16:35,930 --> 00:16:37,420 especially, what is n? 297 00:16:39,950 --> 00:16:44,920 I would like to know how the growth is progressing. 298 00:16:44,920 --> 00:16:50,220 And I'm just taking simple growth law here. 299 00:16:50,220 --> 00:16:50,870 OK. 300 00:16:50,870 --> 00:16:51,960 I would graph it. 301 00:16:51,960 --> 00:16:54,710 I'd get a bunch of points, I put them on a graph, and I 302 00:16:54,710 --> 00:16:56,410 look at the graph. 303 00:16:56,410 --> 00:17:00,080 Now if I just graph these things, if I just graph that 304 00:17:00,080 --> 00:17:10,410 y, here is x and here's y, suppose n is 1.5. 305 00:17:10,410 --> 00:17:13,680 Suppose my growth rate, and this is very 306 00:17:13,680 --> 00:17:17,440 possible, is x to the 1.5. 307 00:17:20,410 --> 00:17:22,460 And a is some number-- 308 00:17:22,460 --> 00:17:23,500 who knows. 309 00:17:23,500 --> 00:17:24,910 Could even be 1. 310 00:17:24,910 --> 00:17:26,160 Suppose a was 1. 311 00:17:30,670 --> 00:17:34,110 So then I'm graphing y as x to the 1.5. 312 00:17:34,110 --> 00:17:36,060 What does that look like? 313 00:17:36,060 --> 00:17:39,890 Well, it looks like that. 314 00:17:39,890 --> 00:17:44,260 The problem is that if the real growth-- 315 00:17:44,260 --> 00:17:46,660 the real good relation-- 316 00:17:46,660 --> 00:17:51,530 see, I would have a few points that might be 317 00:17:51,530 --> 00:17:54,750 close to that curve. 318 00:17:54,750 --> 00:18:01,480 But if I'm looking that curve, I frankly could not tell 1.5 319 00:18:01,480 --> 00:18:04,490 from 1.6 growth rate. 320 00:18:04,490 --> 00:18:07,600 The truth is, I couldn't tell it from 2. 321 00:18:07,600 --> 00:18:11,290 I couldn't tell what the actual growth rate is from my 322 00:18:11,290 --> 00:18:14,630 graph, which has a little error, so I'm not too sure. 323 00:18:14,630 --> 00:18:18,450 And the point is x to the 1.5 and x to the 2 would be all-- 324 00:18:18,450 --> 00:18:22,060 If I sketch the graph, it would look like that. 325 00:18:22,060 --> 00:18:25,460 But go to the log scale. 326 00:18:25,460 --> 00:18:28,090 Go to a log log graph. 327 00:18:28,090 --> 00:18:31,220 So I'm going to take logs of both sides, and 328 00:18:31,220 --> 00:18:33,890 look and plot that. 329 00:18:33,890 --> 00:18:37,180 So I take the logs of both sides, so I take the log of my 330 00:18:37,180 --> 00:18:42,980 outputs y, and now this is a product of that times that. 331 00:18:42,980 --> 00:18:45,030 What's the rule for logarithms? 332 00:18:45,030 --> 00:18:49,590 Add logarithms. So this would be log a plus 333 00:18:49,590 --> 00:18:53,990 log of x to the nth. 334 00:18:53,990 --> 00:18:58,370 But now what's the log of x to the nth? 335 00:18:58,370 --> 00:19:00,570 Beautiful again. 336 00:19:00,570 --> 00:19:05,630 This is x times x times x n times. 337 00:19:05,630 --> 00:19:08,000 At least of n is an integer. 338 00:19:08,000 --> 00:19:11,170 Think of it as x multiplied by itself n times. 339 00:19:11,170 --> 00:19:15,400 When I take the logarithm, I add n times. 340 00:19:15,400 --> 00:19:21,550 Log of x to the nth is n log x. 341 00:19:21,550 --> 00:19:26,980 Now that, let me graph that now. 342 00:19:26,980 --> 00:19:29,280 This is now a log picture. 343 00:19:29,280 --> 00:19:38,520 So I'm graphing log y against log x, which was the whole 344 00:19:38,520 --> 00:19:43,450 point of my log scale, to think of doing this. 345 00:19:43,450 --> 00:19:53,300 And what kind of a curve will I see from this equation on 346 00:19:53,300 --> 00:19:54,550 this graph paper? 347 00:19:56,920 --> 00:19:58,170 A straight line. 348 00:20:00,490 --> 00:20:04,620 That is some constants plus some slope. 349 00:20:04,620 --> 00:20:10,070 n will be the slope times the x. 350 00:20:10,070 --> 00:20:17,360 It's like capital Y is capital A plus n times 351 00:20:17,360 --> 00:20:18,900 capital X or something. 352 00:20:18,900 --> 00:20:20,910 But better for me to write log, so we 353 00:20:20,910 --> 00:20:22,190 remember what it is. 354 00:20:22,190 --> 00:20:26,890 So on this paper, suppose-- 355 00:20:26,890 --> 00:20:30,430 I did the example x to the 1.5. 356 00:20:30,430 --> 00:20:31,850 OK. 357 00:20:31,850 --> 00:20:37,820 So in this example, a is 1 and n is 1.5. 358 00:20:37,820 --> 00:20:40,090 So what would my points look like here? 359 00:20:40,090 --> 00:20:46,260 Now remember, I should really allow negative logarithms. 360 00:20:46,260 --> 00:20:50,450 Because this is the point, right? 361 00:20:50,450 --> 00:20:52,270 This is x equals 1 here. 362 00:20:52,270 --> 00:20:54,565 The log is 0, but the number is 1. 363 00:21:00,750 --> 00:21:00,890 Ha, OK. 364 00:21:00,890 --> 00:21:05,090 So when the log is 0, you see, it's going to 365 00:21:05,090 --> 00:21:07,770 be a straight line. 366 00:21:07,770 --> 00:21:12,460 And actually, when I took a to be 1, its logarithm will be 0. 367 00:21:12,460 --> 00:21:14,560 The line would go right through there. 368 00:21:14,560 --> 00:21:17,230 It would have a slope of 1 and 1/2. 369 00:21:17,230 --> 00:21:20,240 My points will be really close to line. 370 00:21:20,240 --> 00:21:25,280 I measure out, if I go out a distance 1, then I go up a 371 00:21:25,280 --> 00:21:26,458 distance 1.5. 372 00:21:26,458 --> 00:21:26,536 Right? 373 00:21:26,536 --> 00:21:27,786 Up 1.5. 374 00:21:32,880 --> 00:21:36,720 When I go across by 1 on the log picture, it 375 00:21:36,720 --> 00:21:38,160 could be down here. 376 00:21:38,160 --> 00:21:42,540 My numbers could be smaller or larger. 377 00:21:42,540 --> 00:21:43,470 A straight line. 378 00:21:43,470 --> 00:21:47,280 I can get out a ruler and estimate the slope far more 379 00:21:47,280 --> 00:21:55,970 accurately than I could hear with a lot more software. 380 00:21:55,970 --> 00:21:56,075 OK. 381 00:21:56,075 --> 00:22:03,500 So that's an important, very important instance in which we 382 00:22:03,500 --> 00:22:06,180 wonder what the rate of growth is, and the 383 00:22:06,180 --> 00:22:08,460 graph shows it to us. 384 00:22:08,460 --> 00:22:12,290 But just make a little point that I've put some points 385 00:22:12,290 --> 00:22:18,420 here, like near a line, and that raises another graph 386 00:22:18,420 --> 00:22:21,260 question of very great importance. 387 00:22:21,260 --> 00:22:25,890 Suppose you have some experiments that put points 388 00:22:25,890 --> 00:22:29,420 close to a line, but not right on a line. 389 00:22:29,420 --> 00:22:32,700 You want to fit a line close to them. 390 00:22:32,700 --> 00:22:37,950 You want to fit the best line to the experimental points. 391 00:22:37,950 --> 00:22:40,830 How do you fit a straight line? 392 00:22:40,830 --> 00:22:42,220 That's an important thing. 393 00:22:42,220 --> 00:22:48,040 And let me save that for a future chance, because I want 394 00:22:48,040 --> 00:22:49,740 to tell you about it. 395 00:22:49,740 --> 00:22:52,460 The best, the standard way is what's 396 00:22:52,460 --> 00:22:54,480 called the least squares. 397 00:22:54,480 --> 00:22:57,520 So least squares is a very important application. 398 00:22:57,520 --> 00:23:02,970 And the best line, it turns out, is a calculus problem. 399 00:23:02,970 --> 00:23:05,580 So for the moment let's pretend they're 400 00:23:05,580 --> 00:23:07,570 right on the line. 401 00:23:07,570 --> 00:23:13,100 Its slope, which we easily find, tells us this number. 402 00:23:13,100 --> 00:23:16,780 May I mention one other behavior? 403 00:23:16,780 --> 00:23:19,190 So another possibility. 404 00:23:19,190 --> 00:23:24,320 If y is not growing polynomially, but suppose y is 405 00:23:24,320 --> 00:23:28,560 growing exponentially-- 406 00:23:28,560 --> 00:23:30,460 I'll just put it here, because it's not 407 00:23:30,460 --> 00:23:32,290 going to be a big deal. 408 00:23:32,290 --> 00:23:35,410 y is some-- 409 00:23:35,410 --> 00:23:41,250 call it b, e to the c x. 410 00:23:41,250 --> 00:23:42,830 So that's a different type of growth. 411 00:23:42,830 --> 00:23:46,000 That's the big part of the today's lecture, is to say, 412 00:23:46,000 --> 00:23:48,510 this is a quite different growth. 413 00:23:48,510 --> 00:23:50,930 But it would be equally hard-- 414 00:23:50,930 --> 00:23:52,480 or even harder-- 415 00:23:52,480 --> 00:23:57,230 to find this growth rate c from an ordinary graph. 416 00:23:57,230 --> 00:24:00,530 The graph would take off even faster than this one. 417 00:24:00,530 --> 00:24:03,130 You couldn't see what's happening. 418 00:24:03,130 --> 00:24:09,550 The good idea is, take logarithms. But what do we 419 00:24:09,550 --> 00:24:10,610 want to do? 420 00:24:10,610 --> 00:24:12,950 We'll take the logarithm of y-- 421 00:24:12,950 --> 00:24:15,930 log y, as before-- 422 00:24:15,930 --> 00:24:24,590 will be the log of B plus the log of e to the cx. 423 00:24:24,590 --> 00:24:27,700 Oh, maybe I should have made this 10 to the cx, just to 424 00:24:27,700 --> 00:24:29,640 make it all-- 425 00:24:29,640 --> 00:24:31,850 instead of the e, I could use the 10. 426 00:24:31,850 --> 00:24:34,040 Whatever. 427 00:24:34,040 --> 00:24:37,540 Because I've been talking about logarithms to the base 428 00:24:37,540 --> 00:24:41,260 10, so let me use the powers of 10 here. 429 00:24:41,260 --> 00:24:45,790 What's the logarithm of 10 to the cx? 430 00:24:45,790 --> 00:24:50,190 When the base is 10, the logarithm is the exponent. 431 00:24:50,190 --> 00:24:52,770 c times x. 432 00:24:52,770 --> 00:24:54,450 So what am I seeing in this equation? 433 00:24:57,790 --> 00:25:01,370 That's an equation when I've taken logarithms, my big 434 00:25:01,370 --> 00:25:03,950 numbers become reasonable. 435 00:25:03,950 --> 00:25:08,060 And also, very small numbers become reasonable. 436 00:25:08,060 --> 00:25:11,990 And I get a straight line again. 437 00:25:11,990 --> 00:25:14,170 I get a straight line. 438 00:25:14,170 --> 00:25:20,640 But it's not in this log paper. 439 00:25:20,640 --> 00:25:25,290 The logarithm of y, the y-axis, the vertical axis, is 440 00:25:25,290 --> 00:25:27,630 still log scale. 441 00:25:27,630 --> 00:25:30,130 But you see it's ordinary x there now. 442 00:25:30,130 --> 00:25:33,440 So I don't use log x for this one. 443 00:25:33,440 --> 00:25:35,510 Just ordinary x. 444 00:25:35,510 --> 00:25:37,780 It's semi log paper. 445 00:25:37,780 --> 00:25:41,060 Logarithm in the in the vertical direction, ordinary 446 00:25:41,060 --> 00:25:42,970 in the x direction. 447 00:25:42,970 --> 00:25:44,720 OK. 448 00:25:44,720 --> 00:25:45,950 Good. 449 00:25:45,950 --> 00:25:52,380 Now I just want to add one sort of example. 450 00:25:52,380 --> 00:25:56,083 Because it's quite important and also quite practical. 451 00:25:56,083 --> 00:25:57,333 May I tell you about-- 452 00:26:02,840 --> 00:26:06,520 Let me ask you the question, and see if you get an idea. 453 00:26:06,520 --> 00:26:09,150 Because this is like basic to calculus. 454 00:26:09,150 --> 00:26:12,880 Let me talk about-- 455 00:26:12,880 --> 00:26:15,860 this e will stand for error. 456 00:26:15,860 --> 00:26:19,075 Error e. 457 00:26:19,075 --> 00:26:21,760 And what error am I talking about? 458 00:26:21,760 --> 00:26:26,120 I'm talking about the error as the difference between the 459 00:26:26,120 --> 00:26:28,490 derivative-- 460 00:26:28,490 --> 00:26:33,920 I have some function f of x. 461 00:26:33,920 --> 00:26:36,000 And there's its derivative. 462 00:26:36,000 --> 00:26:44,420 And I compare that with delta f over delta x. 463 00:26:44,420 --> 00:26:47,230 So what do I know? 464 00:26:47,230 --> 00:26:52,240 I know that as this is a function of delta x, I'm 465 00:26:52,240 --> 00:26:58,410 comparing the instant slope versus the average slope over 466 00:26:58,410 --> 00:27:02,040 a distance delta x. 467 00:27:02,040 --> 00:27:05,680 So it's not 0, right? 468 00:27:05,680 --> 00:27:10,040 This one is a finite movement. 469 00:27:10,040 --> 00:27:15,050 Delta x produces a finite moment delta f. 470 00:27:15,050 --> 00:27:20,180 As delta x goes to 0, that does approach this. 471 00:27:20,180 --> 00:27:22,720 So here's my question. 472 00:27:22,720 --> 00:27:27,990 My question is, this is approximately some constant 473 00:27:27,990 --> 00:27:32,680 times delta x to some power. 474 00:27:32,680 --> 00:27:34,815 And my question is, what is n? 475 00:27:40,910 --> 00:27:41,940 How close? 476 00:27:41,940 --> 00:27:47,340 What's a rough estimate of how near the delta f over delta x 477 00:27:47,340 --> 00:27:49,860 is to the actual derivative? 478 00:27:49,860 --> 00:27:51,180 OK. 479 00:27:51,180 --> 00:27:56,240 So I have to tell you what I meant by delta f over delta x. 480 00:27:56,240 --> 00:28:05,040 I meant what you also meant, f at x plus delta x minus f at x 481 00:28:05,040 --> 00:28:07,850 divided by delta x. 482 00:28:07,850 --> 00:28:10,720 In other words, that's the familiar delta f. 483 00:28:10,720 --> 00:28:14,470 Moving forward from x, I would call that a forward 484 00:28:14,470 --> 00:28:16,930 difference, a forward delta f. 485 00:28:16,930 --> 00:28:21,180 Because I'm starting at x, and I think of delta x as moving 486 00:28:21,180 --> 00:28:22,940 me a little bit forward. 487 00:28:22,940 --> 00:28:26,240 So I get the delta f, I divide by the delta x, and that's 488 00:28:26,240 --> 00:28:27,560 what this thing means. 489 00:28:27,560 --> 00:28:29,040 And do you know what n is? 490 00:28:33,620 --> 00:28:37,590 Let me connect it to my pictures. 491 00:28:37,590 --> 00:28:42,345 If I tried to graph this, I'd have a graph. 492 00:28:44,850 --> 00:28:45,690 You know. 493 00:28:45,690 --> 00:28:49,720 Here's my delta x and here's my e. 494 00:28:49,720 --> 00:28:53,750 This difference says delta x goes to 0, it goes to 0. 495 00:28:53,750 --> 00:28:57,370 You know, if delta x is small, e is small. 496 00:28:57,370 --> 00:29:00,390 If I divide delta x by 10, e divides by something. 497 00:29:04,060 --> 00:29:06,320 I don't even know if you see it on the camera. 498 00:29:06,320 --> 00:29:08,800 The graph has gone into a-- 499 00:29:08,800 --> 00:29:11,210 well, a black hole, or a chalk hole, or a 500 00:29:11,210 --> 00:29:12,740 white hole, or something. 501 00:29:12,740 --> 00:29:16,550 It's just completely invisible. 502 00:29:16,550 --> 00:29:18,940 I can't see the slope of this thing. 503 00:29:21,570 --> 00:29:26,720 But if I did it on log log paper, I'd see it clearly. 504 00:29:26,720 --> 00:29:30,420 And the answer would be 1. 505 00:29:30,420 --> 00:29:37,100 The error, the difference between derivative and average 506 00:29:37,100 --> 00:29:43,210 slope, goes like delta x to the first power. 507 00:29:43,210 --> 00:29:47,680 And then we can see later where that 1 comes from, and 508 00:29:47,680 --> 00:29:49,800 we can see where that a is. 509 00:29:49,800 --> 00:29:52,690 It's all in Taylor series. 510 00:29:52,690 --> 00:29:57,150 But here's my practical point. 511 00:29:57,150 --> 00:30:00,820 There is a much better delta f than this one. 512 00:30:00,820 --> 00:30:04,290 A much better delta f over delta x. 513 00:30:04,290 --> 00:30:09,090 An average slope that's much more accurate, and that in 514 00:30:09,090 --> 00:30:11,730 calculation I would always use. 515 00:30:11,730 --> 00:30:14,900 And the trouble with this one is, it's lopsided. 516 00:30:14,900 --> 00:30:15,920 It's one-sided. 517 00:30:15,920 --> 00:30:18,160 I only went forward. 518 00:30:18,160 --> 00:30:22,200 Or if delta x is negative, I'm only going backwards. 519 00:30:22,200 --> 00:30:25,920 And it turns out that the average of forward and 520 00:30:25,920 --> 00:30:31,200 backward is like centered at difference. 521 00:30:31,200 --> 00:30:34,070 So let me tell you a center difference. f at 522 00:30:34,070 --> 00:30:36,230 x plus delta x. 523 00:30:36,230 --> 00:30:40,690 So look a little forward, but take the difference from 524 00:30:40,690 --> 00:30:41,940 looking a little backward. 525 00:30:44,410 --> 00:30:48,380 That would be my change in f. 526 00:30:48,380 --> 00:30:54,720 But now what do I divide by to get a reasonable slope? 527 00:30:54,720 --> 00:31:00,050 Well, this is the change in f going from minus delta x-- 528 00:31:00,050 --> 00:31:03,370 delta x to the left of the point to delta x to the right 529 00:31:03,370 --> 00:31:04,350 of the point. 530 00:31:04,350 --> 00:31:09,840 The real movement there in the x-axis was a movement 531 00:31:09,840 --> 00:31:11,090 of two delta xs. 532 00:31:14,580 --> 00:31:17,200 So I would call this a center difference. 533 00:31:17,200 --> 00:31:19,099 Can I write that word "centered" down? 534 00:31:24,730 --> 00:31:29,810 And if I use that, which is a lot smarter if I'm practically 535 00:31:29,810 --> 00:31:33,050 wanting to get pictures, then what happens? 536 00:31:33,050 --> 00:31:37,350 So if this is now instead of this, instead of choosing this 537 00:31:37,350 --> 00:31:45,200 lopsided, simple, familiar but not that great difference, if 538 00:31:45,200 --> 00:31:53,710 I go for this one, the answer is, n changes to 2. 539 00:31:53,710 --> 00:31:56,350 n is 2 for this one. 540 00:31:56,350 --> 00:32:03,960 The accuracy is way, way better for center differences. 541 00:32:03,960 --> 00:32:09,510 And the point about the log graphs is, if I plot those 542 00:32:09,510 --> 00:32:14,170 points on the graph I would see that slope of 2 in the log 543 00:32:14,170 --> 00:32:17,460 log graph, it would be again-- 544 00:32:17,460 --> 00:32:22,780 in ordinary graph, it would become invisible 545 00:32:22,780 --> 00:32:24,690 as delta x got small. 546 00:32:24,690 --> 00:32:30,120 But on a log scale, I'd see it perfectly. 547 00:32:30,120 --> 00:32:31,300 OK. 548 00:32:31,300 --> 00:32:35,650 Some practical uses of logarithms. Now that we no 549 00:32:35,650 --> 00:32:39,550 longer use slide rules, this is what we do. 550 00:32:39,550 --> 00:32:41,650 Thanks. 551 00:32:41,650 --> 00:32:43,410 NARRATOR: This has been a production of MIT 552 00:32:43,410 --> 00:32:45,800 OpenCourseWare and Gilbert Strang. 553 00:32:45,800 --> 00:32:48,080 Funding for this video was provided by the Lord 554 00:32:48,080 --> 00:32:49,290 Foundation. 555 00:32:49,290 --> 00:32:52,420 To help OCW continue to provide free and open access 556 00:32:52,420 --> 00:32:55,500 to MIT courses, please make a donation at 557 00:32:55,500 --> 00:32:57,060 ocw.mit.edu/donate.