WEBVTT 00:00:07.600 --> 00:00:07.860 PROFESSOR: OK. 00:00:07.860 --> 00:00:08.350 Hi. 00:00:08.350 --> 00:00:13.620 I thought I'd give a short lecture about how logarithms 00:00:13.620 --> 00:00:15.230 are actually used. 00:00:15.230 --> 00:00:17.760 So a little bit practical. 00:00:17.760 --> 00:00:22.750 And also, it naturally comes in, how quickly 00:00:22.750 --> 00:00:24.720 do functions grow? 00:00:24.720 --> 00:00:28.350 Which functions grow faster than others? 00:00:28.350 --> 00:00:33.700 And I made a list of a bunch of functions that 00:00:33.700 --> 00:00:35.290 we see all the time. 00:00:35.290 --> 00:00:37.520 Linear growth. 00:00:37.520 --> 00:00:41.440 Just, the function goes up along the straight line. 00:00:41.440 --> 00:00:46.230 Proportional to x, linear could have been a c times x, 00:00:46.230 --> 00:00:48.240 still linear. 00:00:48.240 --> 00:00:54.990 Here that's called polynomial growth, like some power of x. 00:00:54.990 --> 00:00:58.080 Here is faster growth. 00:00:58.080 --> 00:01:02.100 We introduced e to the x, and I'll take this chance to bring 00:01:02.100 --> 00:01:05.300 in 2 to the x and 10 to the x. 00:01:05.300 --> 00:01:08.180 Especially 10 to the x, because that'll lead us to 00:01:08.180 --> 00:01:16.110 logarithms to base 10, and those are handy in practice. 00:01:16.110 --> 00:01:18.040 So that's exponential growth. 00:01:18.040 --> 00:01:21.680 And here are some that grow faster still. 00:01:21.680 --> 00:01:27.655 x factorial, n factorial grows really fast. And n to the nth 00:01:27.655 --> 00:01:31.810 or x to the xth is a function that grows still faster. 00:01:31.810 --> 00:01:34.960 And of course, we could cook up a function the 00:01:34.960 --> 00:01:36.140 grew faster than that. 00:01:36.140 --> 00:01:40.680 X to the x to the x power would really just take off. 00:01:40.680 --> 00:01:44.180 And we could find functions that grow more slowly. 00:01:44.180 --> 00:01:50.770 But let's just take these and let x be 1000. 00:01:50.770 --> 00:01:54.770 Just to have a kind of realistic idea of how these 00:01:54.770 --> 00:01:57.320 compare when x is 1000. 00:01:57.320 --> 00:01:57.870 OK. 00:01:57.870 --> 00:02:00.060 So I'm skipping to c. 00:02:00.060 --> 00:02:02.390 So x will be 1000. 00:02:02.390 --> 00:02:04.360 10 cubed. 00:02:04.360 --> 00:02:11.810 Let me just write it as 10 cubed. 00:02:11.810 --> 00:02:15.940 So x is going to be 1000. 00:02:15.940 --> 00:02:18.430 And because these are big numbers, I'm going to write 00:02:18.430 --> 00:02:21.170 them as powers of 10. 00:02:21.170 --> 00:02:22.280 OK. 00:02:22.280 --> 00:02:25.460 so how about 1000 squared? 00:02:25.460 --> 00:02:34.560 10 cubed squared will be 10 to the sixth. 00:02:34.560 --> 00:02:40.030 1000 cubed, we're up to 10 to the ninth. 00:02:40.030 --> 00:02:41.500 And onwards. 00:02:41.500 --> 00:02:45.420 Like, this is where the economists are working. 00:02:45.420 --> 00:02:49.460 The national debt is in this range. 00:02:49.460 --> 00:02:49.840 OK. 00:02:49.840 --> 00:02:54.070 Now fortunately, it's not in this range. 00:02:54.070 --> 00:02:56.640 2 to the thousandth power. 00:02:56.640 --> 00:02:59.620 And if I want to be able to compare it, I'll write that 00:02:59.620 --> 00:03:03.540 approximately as 10 to-- 00:03:03.540 --> 00:03:07.710 well, if it's 2 to the thousandth power, it'll be 10 00:03:07.710 --> 00:03:09.850 to a smaller power. 00:03:09.850 --> 00:03:16.980 And 300 is pretty close for 2 to the thousandth. 00:03:16.980 --> 00:03:21.150 Then e to the thousandth, that's going to be bigger than 00:03:21.150 --> 00:03:24.890 2. e is 2.7 et cetera. 00:03:24.890 --> 00:03:27.300 This is more like 10 to the-- 00:03:27.300 --> 00:03:33.770 I think this is right-- about 434, maybe. 00:03:33.770 --> 00:03:36.410 And 10 to the thousandth-- 00:03:36.410 --> 00:03:37.960 well, I can write that right in. 00:03:37.960 --> 00:03:42.380 10 to the thousandth when x is 1000. 00:03:42.380 --> 00:03:43.370 OK. 00:03:43.370 --> 00:03:46.000 So that's the one that is exactly right. 00:03:46.000 --> 00:03:51.540 And also, I could write in 1000 to the thousandth power. 00:03:51.540 --> 00:03:54.260 What power of 10 will this be? 00:03:54.260 --> 00:03:56.470 10 to the what? 00:03:56.470 --> 00:04:00.730 1000 to the thousandth power, I think, is 10 to the three 00:04:00.730 --> 00:04:02.690 thousandth. 00:04:02.690 --> 00:04:04.850 Why do I think that? 00:04:04.850 --> 00:04:10.920 Because 1000 itself is 10 times 10 times 10. 00:04:10.920 --> 00:04:13.180 Three of them, right? 00:04:13.180 --> 00:04:17.779 And then we do that 1000 times, so we have a string of 00:04:17.779 --> 00:04:21.620 3000 10s multiplying each other. 00:04:21.620 --> 00:04:23.960 And that's what 10 to the three thousandth is. 00:04:23.960 --> 00:04:29.120 And you might wonder about a thousand factorial. 00:04:29.120 --> 00:04:31.695 Let me make the rough estimate. 00:04:34.220 --> 00:04:38.010 A big number in factorial, order of magnitude, is 00:04:38.010 --> 00:04:41.220 something like, it doesn't grow as fast as this, because 00:04:41.220 --> 00:04:45.480 this is x times x minus 1 times x minus 2. 00:04:45.480 --> 00:04:48.990 1000 times 999 times 998. 00:04:48.990 --> 00:04:52.710 So we're not repeating 1000 every time. 00:04:52.710 --> 00:04:54.150 And the difference-- 00:04:54.150 --> 00:04:59.630 it turns out that this number divided by this number, x to 00:04:59.630 --> 00:05:03.130 the x over e to the x, is the right 00:05:03.130 --> 00:05:05.700 general picture for factorial. 00:05:05.700 --> 00:05:10.490 So that would be, if I divide 10 to the 3000 by 10 and this 00:05:10.490 --> 00:05:11.550 power, what do I do? 00:05:11.550 --> 00:05:14.320 In a division, I do a subtraction of exponents, 00:05:14.320 --> 00:05:18.530 because I have that many fewer 10s multiplying each other. 00:05:18.530 --> 00:05:22.300 So I think it would be 3000, but I don't want the full 00:05:22.300 --> 00:05:27.530 3000, because I take away e to the thousandth, 434 of them. 00:05:27.530 --> 00:05:28.520 So that's about-- 00:05:28.520 --> 00:05:34.750 2566 is close enough, anyway. 00:05:34.750 --> 00:05:35.190 OK. 00:05:35.190 --> 00:05:36.440 Giant numbers. 00:05:36.440 --> 00:05:37.320 Giant numbers. 00:05:37.320 --> 00:05:41.970 And of course you saw that I didn't write it out with 1 and 00:05:41.970 --> 00:05:43.375 3000, or whatever, zeros. 00:05:43.375 --> 00:05:43.530 Hopeless. 00:05:43.530 --> 00:05:44.780 OK. 00:05:46.760 --> 00:05:51.800 In other words, it's the exponent that gives me 00:05:51.800 --> 00:05:53.350 something I can really work with. 00:05:53.350 --> 00:05:55.850 And the exponent is the logarithm. 00:05:55.850 --> 00:05:57.150 That's what logarithms are. 00:05:57.150 --> 00:05:58.690 They are the exponents. 00:05:58.690 --> 00:06:05.190 And when they're the exponent with a 10, I call 10 the base. 00:06:05.190 --> 00:06:09.290 And I'm speaking about logarithms to the base 10. 00:06:09.290 --> 00:06:11.850 Can I just copy those numbers again? 00:06:11.850 --> 00:06:14.250 And then I want to write their logarithms. Because it's the 00:06:14.250 --> 00:06:21.210 logarithms that kind of remain reasonable-looking numbers but 00:06:21.210 --> 00:06:25.180 tell you very nicely what's growing fast. 00:06:25.180 --> 00:06:27.340 So let me write out again. 00:06:27.340 --> 00:06:33.380 10 cubed, 10 sixth, 10 to the ninth is polynomial growth 00:06:33.380 --> 00:06:35.670 starting with the first power. 00:06:35.670 --> 00:06:38.800 Then I'll write down 10 to the three hundredth, 00:06:38.800 --> 00:06:40.280 approximately. 00:06:40.280 --> 00:06:43.920 10 to the 434, I think, is about right. 00:06:43.920 --> 00:06:46.760 And then 10 to the 1000. 00:06:46.760 --> 00:06:53.910 And then I had 10 to the 2566 as something, roughly 1000 00:06:53.910 --> 00:06:59.370 factorial, and then 10 to the 3000. 00:06:59.370 --> 00:06:59.660 OK. 00:06:59.660 --> 00:07:02.120 I just copied those numbers again. 00:07:02.120 --> 00:07:06.660 And now I plan to take their logarithms. I can see what's 00:07:06.660 --> 00:07:10.360 happening with logarithms. The logarithm of 10 00:07:10.360 --> 00:07:12.750 to the ninth is-- 00:07:12.750 --> 00:07:14.090 if the base is 10-- 00:07:14.090 --> 00:07:19.070 the logarithm of 10 to the ninth is the nine. 00:07:19.070 --> 00:07:21.170 This has logarithm 6. 00:07:21.170 --> 00:07:23.000 This has logarithm 3. 00:07:23.000 --> 00:07:24.700 So you see-- 00:07:24.700 --> 00:07:24.970 well. 00:07:24.970 --> 00:07:27.220 If we took the logarithm of the national debt, it wouldn't 00:07:27.220 --> 00:07:29.250 look too serious. 00:07:29.250 --> 00:07:34.540 It would just be up around 9 moving toward 10. 00:07:34.540 --> 00:07:39.510 But what I'm using it for here is to get some 00:07:39.510 --> 00:07:41.990 reasonable way to see-- 00:07:41.990 --> 00:07:42.840 300. 00:07:42.840 --> 00:07:44.355 Of course, that's big. 00:07:44.355 --> 00:07:46.370 For a logarithm, that's a very big number. 00:07:46.370 --> 00:07:49.190 434, 1000. 00:07:49.190 --> 00:07:51.310 These are climbing up. 00:07:51.310 --> 00:07:55.910 2566 and 3000. 00:07:55.910 --> 00:07:58.280 OK. 00:07:58.280 --> 00:07:59.530 So these are the logs. 00:08:02.270 --> 00:08:04.100 Just to repeat. 00:08:04.100 --> 00:08:08.200 If I wanted this growth, this list of functions by how fast 00:08:08.200 --> 00:08:11.440 they grow, where would log x appear 00:08:11.440 --> 00:08:13.580 in my list of functions? 00:08:13.580 --> 00:08:16.310 It would be way at the left end. 00:08:16.310 --> 00:08:17.840 Slower than x. 00:08:17.840 --> 00:08:18.980 Much slower than x. 00:08:18.980 --> 00:08:23.040 Log x grows very slowly, as we see here. 00:08:23.040 --> 00:08:25.590 And then if you wanted one that really grew slowly, it 00:08:25.590 --> 00:08:28.080 would be log of log x. 00:08:28.080 --> 00:08:30.180 That creeps along. 00:08:30.180 --> 00:08:32.590 Eventually gets to-- 00:08:32.590 --> 00:08:33.940 passes any number. 00:08:33.940 --> 00:08:36.400 But x has to be enormous. 00:08:36.400 --> 00:08:40.946 And one more little comment before I begin to use some 00:08:40.946 --> 00:08:43.120 things graphically. 00:08:43.120 --> 00:08:48.445 Because that's the other part of this talk, is log-- 00:08:48.445 --> 00:08:49.850 the graphs. 00:08:49.850 --> 00:08:51.810 Using logarithms in graphs. 00:08:51.810 --> 00:08:53.240 A little point. 00:08:53.240 --> 00:08:57.340 You might ask, what about functions that decay? 00:08:57.340 --> 00:09:00.850 What would be the corresponding functions here 00:09:00.850 --> 00:09:01.890 that decay? 00:09:01.890 --> 00:09:03.460 Let me write them here. 00:09:03.460 --> 00:09:04.710 Decay. 00:09:06.700 --> 00:09:09.610 By that I mean, headed for 0 instead 00:09:09.610 --> 00:09:11.110 of headed for infinity. 00:09:11.110 --> 00:09:17.110 Well, 1 over x, 1 over x squared, 1 over x cubed. 00:09:20.300 --> 00:09:27.080 Those functions go to 0 faster and faster. 00:09:27.080 --> 00:09:29.410 Now, what about these? 00:09:29.410 --> 00:09:33.190 The next list would be 1 over-- 00:09:33.190 --> 00:09:36.070 I'm dividing, but 1 over 2 to the x. 00:09:36.070 --> 00:09:37.370 1 over e to the x. 00:09:37.370 --> 00:09:38.970 Can I write that in a better way? 00:09:38.970 --> 00:09:40.880 e to the minus x. 00:09:40.880 --> 00:09:45.980 1 over 10 to the x. 00:09:45.980 --> 00:09:49.290 Those are going to 0 like crazy. 00:09:49.290 --> 00:09:52.730 And of course, if I keep going, even worse. 00:09:52.730 --> 00:09:59.430 So like, x to the minus x power would be really small. 00:09:59.430 --> 00:10:05.220 So my point is just that we have a scale here that not 00:10:05.220 --> 00:10:09.720 only gives us a handle of how to deal with things that are 00:10:09.720 --> 00:10:13.130 growing very fast, but also things that are going to 0 00:10:13.130 --> 00:10:16.920 very fast. The other, the negative logarithms. The 00:10:16.920 --> 00:10:20.590 logarithms of these things would be minus 3, minus 6, 00:10:20.590 --> 00:10:25.260 minus 9 and so on, if I divide by one. 00:10:25.260 --> 00:10:26.120 Good. 00:10:26.120 --> 00:10:26.630 All right. 00:10:26.630 --> 00:10:29.610 So that suggests the idea. 00:10:29.610 --> 00:10:36.000 Now I want to introduce the idea of a log scale. 00:10:36.000 --> 00:10:48.475 So I'm just going to think of a usual straight line, on 00:10:48.475 --> 00:10:54.290 which we usually mark out 0, 1, 2, 3, minus 1, minus 2. 00:10:54.290 --> 00:11:04.170 But on this log scale, the center point, the 0, I'm 00:11:04.170 --> 00:11:08.140 really graphing the logarithm of x instead of x. 00:11:08.140 --> 00:11:09.250 That's the point. 00:11:09.250 --> 00:11:12.640 That in this log scale, what I'm picturing 00:11:12.640 --> 00:11:13.890 along here will be-- 00:11:16.210 --> 00:11:22.680 this number will be 10 to the 0 power, which is 1. 00:11:22.680 --> 00:11:25.270 The next one will be 10. 00:11:25.270 --> 00:11:27.180 The next one will be 100. 00:11:27.180 --> 00:11:29.890 The next one will be 1000. 00:11:29.890 --> 00:11:34.850 So you see, within this picture-- 00:11:34.850 --> 00:11:37.440 on a graph that we could draw and look at 00:11:37.440 --> 00:11:39.370 on a printed page-- 00:11:39.370 --> 00:11:44.000 we can get big numbers by going from the ordinary 1, 2, 00:11:44.000 --> 00:11:48.900 3 scale to the log scale, which puts these points in 00:11:48.900 --> 00:11:49.430 this order. 00:11:49.430 --> 00:11:51.150 And let me put some of the other ones. 00:11:51.150 --> 00:11:54.080 Now, what one point goes there? 00:11:54.080 --> 00:11:55.860 1/10. 00:11:55.860 --> 00:12:00.640 Every time I go that far, I'm multiplying by 10. 00:12:00.640 --> 00:12:02.850 When I go this way, I'm dividing by 10. 00:12:02.850 --> 00:12:07.970 Up there, this is the number 1/10, which is the same as 10 00:12:07.970 --> 00:12:10.720 to the minus 1 power, right? 00:12:10.720 --> 00:12:14.850 Here is one hundredth. 00:12:14.850 --> 00:12:18.080 Here is one thousandth. 00:12:18.080 --> 00:12:18.610 And so on. 00:12:18.610 --> 00:12:24.820 So this log scale is able to deal with very small numbers 00:12:24.820 --> 00:12:30.540 and very large numbers in a reasonable way. 00:12:30.540 --> 00:12:35.530 And everybody sees the point here that really, what it is 00:12:35.530 --> 00:12:39.170 is the logarithms. So this is 0. 00:12:39.170 --> 00:12:43.550 This is 1, 2, 3, and so on. 00:12:43.550 --> 00:12:46.500 Minus 1, minus 2, minus 3. 00:12:46.500 --> 00:12:51.350 If I'm graphing, really, these are the logarithms of x. 00:12:51.350 --> 00:12:55.470 And I'm doing logs to base 10 again, because that gives us 00:12:55.470 --> 00:12:57.630 nice numbers. 00:12:57.630 --> 00:12:57.705 OK. 00:12:57.705 --> 00:13:00.560 By the way, what's that number? 00:13:00.560 --> 00:13:04.355 What's that number, halfway between there and there? 00:13:08.070 --> 00:13:13.910 It's not halfway between 1 and 10 in the ordinary sense, 00:13:13.910 --> 00:13:15.760 which is whatever, 5 and a half. 00:13:15.760 --> 00:13:17.020 No way. 00:13:17.020 --> 00:13:20.520 Halfway between here is-- 00:13:20.520 --> 00:13:21.580 you know what it will be? 00:13:21.580 --> 00:13:25.870 It'll be square root of 10. 00:13:25.870 --> 00:13:27.840 10 to the 1/2 power. 00:13:27.840 --> 00:13:33.720 The half is here. 00:13:33.720 --> 00:13:36.720 The log is a half, so the number is the 00:13:36.720 --> 00:13:37.490 square root of 10. 00:13:37.490 --> 00:13:40.760 That's about 3, a little more than 3. 00:13:40.760 --> 00:13:45.000 And what would be here, would be 10 to the minus 1/2. 00:13:45.000 --> 00:13:47.450 1 over square root of 10. 00:13:47.450 --> 00:13:49.800 So you see that picture. 00:13:49.800 --> 00:13:53.840 Oh, I have another question, before I use the scales. 00:13:53.840 --> 00:13:56.900 What if I like the powers of 2 better? 00:13:56.900 --> 00:14:00.320 In many cases, we might prefer powers of 2. 00:14:00.320 --> 00:14:04.370 Well, if I plotted the numbers-- 00:14:04.370 --> 00:14:07.620 I'm looking at this log scale. 00:14:07.620 --> 00:14:13.050 And suppose I plot the numbers 1, 2, 4, 8, whatever. 00:14:13.050 --> 00:14:14.720 16. 00:14:14.720 --> 00:14:16.520 What could you tell me about those? 00:14:16.520 --> 00:14:17.890 Well, I know where 1 is. 00:14:17.890 --> 00:14:19.370 It's right there. 00:14:19.370 --> 00:14:20.780 That's a 1. 00:14:20.780 --> 00:14:22.900 Well, two would be a little further over. 00:14:22.900 --> 00:14:26.590 Then 4, then 8 would come before 10, and 16 00:14:26.590 --> 00:14:28.280 would come after 10. 00:14:28.280 --> 00:14:32.020 I pointed there, but 16 would not come there. 00:14:32.020 --> 00:14:37.390 16 would be a lot closer, I think, in here. 00:14:37.390 --> 00:14:43.720 What's the deal with 1, 2, 4, 8, 16 on this log scale? 00:14:43.720 --> 00:14:45.970 They would be equally spaced. 00:14:48.790 --> 00:14:50.350 Of course, the spacing would be 00:14:50.350 --> 00:14:52.810 smaller than the 10 spacing. 00:14:52.810 --> 00:14:58.520 If every time I multiplied by 2, I go the same distance. 00:14:58.520 --> 00:15:01.500 After I'd done it about 10 times-- 00:15:01.500 --> 00:15:05.690 multiplied by 2 10 times-- so that's 2 to the tenth power is 00:15:05.690 --> 00:15:07.060 close to 1000. 00:15:07.060 --> 00:15:11.445 So 10 powers of 2 would bring me pretty near there. 00:15:11.445 --> 00:15:13.760 Anyway. 00:15:13.760 --> 00:15:15.830 And here's one more question. 00:15:15.830 --> 00:15:17.850 Where is 0? 00:15:17.850 --> 00:15:25.140 If my value that I wanted to plot happened to be 0, where 00:15:25.140 --> 00:15:28.530 is it on this graph? 00:15:28.530 --> 00:15:30.670 It's not there. 00:15:30.670 --> 00:15:34.270 You can't plot 0 on a log scale. 00:15:34.270 --> 00:15:37.780 It's way down at the-- 00:15:37.780 --> 00:15:41.620 you know, it's at the minus infinity end of the graph. 00:15:41.620 --> 00:15:48.390 Infinity is up there at that end, and 0 is down here. 00:15:48.390 --> 00:15:49.800 OK. 00:15:49.800 --> 00:15:50.330 Good. 00:15:50.330 --> 00:15:51.930 So can we use that log scale? 00:15:51.930 --> 00:15:53.780 How do we use that log scale? 00:15:53.780 --> 00:15:57.790 Let me give you an idea for what use that 00:15:57.790 --> 00:15:59.710 log scale might be. 00:15:59.710 --> 00:16:01.520 Practical use. 00:16:01.520 --> 00:16:08.600 Suppose I know, or have reason to believe, that my function 00:16:08.600 --> 00:16:17.580 might be of the form y is something times x to the nth. 00:16:17.580 --> 00:16:22.030 I have some quantity y. 00:16:22.030 --> 00:16:25.880 The output when the input is x. 00:16:25.880 --> 00:16:31.450 But I don't know these, that number a. 00:16:31.450 --> 00:16:32.960 So I've done an experiment. 00:16:32.960 --> 00:16:35.930 And I would like to know what is a, and 00:16:35.930 --> 00:16:37.420 especially, what is n? 00:16:39.950 --> 00:16:44.920 I would like to know how the growth is progressing. 00:16:44.920 --> 00:16:50.220 And I'm just taking simple growth law here. 00:16:50.220 --> 00:16:50.870 OK. 00:16:50.870 --> 00:16:51.960 I would graph it. 00:16:51.960 --> 00:16:54.710 I'd get a bunch of points, I put them on a graph, and I 00:16:54.710 --> 00:16:56.410 look at the graph. 00:16:56.410 --> 00:17:00.080 Now if I just graph these things, if I just graph that 00:17:00.080 --> 00:17:10.410 y, here is x and here's y, suppose n is 1.5. 00:17:10.410 --> 00:17:13.680 Suppose my growth rate, and this is very 00:17:13.680 --> 00:17:17.440 possible, is x to the 1.5. 00:17:20.410 --> 00:17:22.460 And a is some number-- 00:17:22.460 --> 00:17:23.500 who knows. 00:17:23.500 --> 00:17:24.910 Could even be 1. 00:17:24.910 --> 00:17:26.160 Suppose a was 1. 00:17:30.670 --> 00:17:34.110 So then I'm graphing y as x to the 1.5. 00:17:34.110 --> 00:17:36.060 What does that look like? 00:17:36.060 --> 00:17:39.890 Well, it looks like that. 00:17:39.890 --> 00:17:44.260 The problem is that if the real growth-- 00:17:44.260 --> 00:17:46.660 the real good relation-- 00:17:46.660 --> 00:17:51.530 see, I would have a few points that might be 00:17:51.530 --> 00:17:54.750 close to that curve. 00:17:54.750 --> 00:18:01.480 But if I'm looking that curve, I frankly could not tell 1.5 00:18:01.480 --> 00:18:04.490 from 1.6 growth rate. 00:18:04.490 --> 00:18:07.600 The truth is, I couldn't tell it from 2. 00:18:07.600 --> 00:18:11.290 I couldn't tell what the actual growth rate is from my 00:18:11.290 --> 00:18:14.630 graph, which has a little error, so I'm not too sure. 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-- 00:18:18.450 --> 00:18:22.060 If I sketch the graph, it would look like that. 00:18:22.060 --> 00:18:25.460 But go to the log scale. 00:18:25.460 --> 00:18:28.090 Go to a log log graph. 00:18:28.090 --> 00:18:31.220 So I'm going to take logs of both sides, and 00:18:31.220 --> 00:18:33.890 look and plot that. 00:18:33.890 --> 00:18:37.180 So I take the logs of both sides, so I take the log of my 00:18:37.180 --> 00:18:42.980 outputs y, and now this is a product of that times that. 00:18:42.980 --> 00:18:45.030 What's the rule for logarithms? 00:18:45.030 --> 00:18:49.590 Add logarithms. So this would be log a plus 00:18:49.590 --> 00:18:53.990 log of x to the nth. 00:18:53.990 --> 00:18:58.370 But now what's the log of x to the nth? 00:18:58.370 --> 00:19:00.570 Beautiful again. 00:19:00.570 --> 00:19:05.630 This is x times x times x n times. 00:19:05.630 --> 00:19:08.000 At least of n is an integer. 00:19:08.000 --> 00:19:11.170 Think of it as x multiplied by itself n times. 00:19:11.170 --> 00:19:15.400 When I take the logarithm, I add n times. 00:19:15.400 --> 00:19:21.550 Log of x to the nth is n log x. 00:19:21.550 --> 00:19:26.980 Now that, let me graph that now. 00:19:26.980 --> 00:19:29.280 This is now a log picture. 00:19:29.280 --> 00:19:38.520 So I'm graphing log y against log x, which was the whole 00:19:38.520 --> 00:19:43.450 point of my log scale, to think of doing this. 00:19:43.450 --> 00:19:53.300 And what kind of a curve will I see from this equation on 00:19:53.300 --> 00:19:54.550 this graph paper? 00:19:56.920 --> 00:19:58.170 A straight line. 00:20:00.490 --> 00:20:04.620 That is some constants plus some slope. 00:20:04.620 --> 00:20:10.070 n will be the slope times the x. 00:20:10.070 --> 00:20:17.360 It's like capital Y is capital A plus n times 00:20:17.360 --> 00:20:18.900 capital X or something. 00:20:18.900 --> 00:20:20.910 But better for me to write log, so we 00:20:20.910 --> 00:20:22.190 remember what it is. 00:20:22.190 --> 00:20:26.890 So on this paper, suppose-- 00:20:26.890 --> 00:20:30.430 I did the example x to the 1.5. 00:20:30.430 --> 00:20:31.850 OK. 00:20:31.850 --> 00:20:37.820 So in this example, a is 1 and n is 1.5. 00:20:37.820 --> 00:20:40.090 So what would my points look like here? 00:20:40.090 --> 00:20:46.260 Now remember, I should really allow negative logarithms. 00:20:46.260 --> 00:20:50.450 Because this is the point, right? 00:20:50.450 --> 00:20:52.270 This is x equals 1 here. 00:20:52.270 --> 00:20:54.565 The log is 0, but the number is 1. 00:21:00.750 --> 00:21:00.890 Ha, OK. 00:21:00.890 --> 00:21:05.090 So when the log is 0, you see, it's going to 00:21:05.090 --> 00:21:07.770 be a straight line. 00:21:07.770 --> 00:21:12.460 And actually, when I took a to be 1, its logarithm will be 0. 00:21:12.460 --> 00:21:14.560 The line would go right through there. 00:21:14.560 --> 00:21:17.230 It would have a slope of 1 and 1/2. 00:21:17.230 --> 00:21:20.240 My points will be really close to line. 00:21:20.240 --> 00:21:25.280 I measure out, if I go out a distance 1, then I go up a 00:21:25.280 --> 00:21:26.458 distance 1.5. 00:21:26.458 --> 00:21:26.536 Right? 00:21:26.536 --> 00:21:27.786 Up 1.5. 00:21:32.880 --> 00:21:36.720 When I go across by 1 on the log picture, it 00:21:36.720 --> 00:21:38.160 could be down here. 00:21:38.160 --> 00:21:42.540 My numbers could be smaller or larger. 00:21:42.540 --> 00:21:43.470 A straight line. 00:21:43.470 --> 00:21:47.280 I can get out a ruler and estimate the slope far more 00:21:47.280 --> 00:21:55.970 accurately than I could hear with a lot more software. 00:21:55.970 --> 00:21:56.075 OK. 00:21:56.075 --> 00:22:03.500 So that's an important, very important instance in which we 00:22:03.500 --> 00:22:06.180 wonder what the rate of growth is, and the 00:22:06.180 --> 00:22:08.460 graph shows it to us. 00:22:08.460 --> 00:22:12.290 But just make a little point that I've put some points 00:22:12.290 --> 00:22:18.420 here, like near a line, and that raises another graph 00:22:18.420 --> 00:22:21.260 question of very great importance. 00:22:21.260 --> 00:22:25.890 Suppose you have some experiments that put points 00:22:25.890 --> 00:22:29.420 close to a line, but not right on a line. 00:22:29.420 --> 00:22:32.700 You want to fit a line close to them. 00:22:32.700 --> 00:22:37.950 You want to fit the best line to the experimental points. 00:22:37.950 --> 00:22:40.830 How do you fit a straight line? 00:22:40.830 --> 00:22:42.220 That's an important thing. 00:22:42.220 --> 00:22:48.040 And let me save that for a future chance, because I want 00:22:48.040 --> 00:22:49.740 to tell you about it. 00:22:49.740 --> 00:22:52.460 The best, the standard way is what's 00:22:52.460 --> 00:22:54.480 called the least squares. 00:22:54.480 --> 00:22:57.520 So least squares is a very important application. 00:22:57.520 --> 00:23:02.970 And the best line, it turns out, is a calculus problem. 00:23:02.970 --> 00:23:05.580 So for the moment let's pretend they're 00:23:05.580 --> 00:23:07.570 right on the line. 00:23:07.570 --> 00:23:13.100 Its slope, which we easily find, tells us this number. 00:23:13.100 --> 00:23:16.780 May I mention one other behavior? 00:23:16.780 --> 00:23:19.190 So another possibility. 00:23:19.190 --> 00:23:24.320 If y is not growing polynomially, but suppose y is 00:23:24.320 --> 00:23:28.560 growing exponentially-- 00:23:28.560 --> 00:23:30.460 I'll just put it here, because it's not 00:23:30.460 --> 00:23:32.290 going to be a big deal. 00:23:32.290 --> 00:23:35.410 y is some-- 00:23:35.410 --> 00:23:41.250 call it b, e to the c x. 00:23:41.250 --> 00:23:42.830 So that's a different type of growth. 00:23:42.830 --> 00:23:46.000 That's the big part of the today's lecture, is to say, 00:23:46.000 --> 00:23:48.510 this is a quite different growth. 00:23:48.510 --> 00:23:50.930 But it would be equally hard-- 00:23:50.930 --> 00:23:52.480 or even harder-- 00:23:52.480 --> 00:23:57.230 to find this growth rate c from an ordinary graph. 00:23:57.230 --> 00:24:00.530 The graph would take off even faster than this one. 00:24:00.530 --> 00:24:03.130 You couldn't see what's happening. 00:24:03.130 --> 00:24:09.550 The good idea is, take logarithms. But what do we 00:24:09.550 --> 00:24:10.610 want to do? 00:24:10.610 --> 00:24:12.950 We'll take the logarithm of y-- 00:24:12.950 --> 00:24:15.930 log y, as before-- 00:24:15.930 --> 00:24:24.590 will be the log of B plus the log of e to the cx. 00:24:24.590 --> 00:24:27.700 Oh, maybe I should have made this 10 to the cx, just to 00:24:27.700 --> 00:24:29.640 make it all-- 00:24:29.640 --> 00:24:31.850 instead of the e, I could use the 10. 00:24:31.850 --> 00:24:34.040 Whatever. 00:24:34.040 --> 00:24:37.540 Because I've been talking about logarithms to the base 00:24:37.540 --> 00:24:41.260 10, so let me use the powers of 10 here. 00:24:41.260 --> 00:24:45.790 What's the logarithm of 10 to the cx? 00:24:45.790 --> 00:24:50.190 When the base is 10, the logarithm is the exponent. 00:24:50.190 --> 00:24:52.770 c times x. 00:24:52.770 --> 00:24:54.450 So what am I seeing in this equation? 00:24:57.790 --> 00:25:01.370 That's an equation when I've taken logarithms, my big 00:25:01.370 --> 00:25:03.950 numbers become reasonable. 00:25:03.950 --> 00:25:08.060 And also, very small numbers become reasonable. 00:25:08.060 --> 00:25:11.990 And I get a straight line again. 00:25:11.990 --> 00:25:14.170 I get a straight line. 00:25:14.170 --> 00:25:20.640 But it's not in this log paper. 00:25:20.640 --> 00:25:25.290 The logarithm of y, the y-axis, the vertical axis, is 00:25:25.290 --> 00:25:27.630 still log scale. 00:25:27.630 --> 00:25:30.130 But you see it's ordinary x there now. 00:25:30.130 --> 00:25:33.440 So I don't use log x for this one. 00:25:33.440 --> 00:25:35.510 Just ordinary x. 00:25:35.510 --> 00:25:37.780 It's semi log paper. 00:25:37.780 --> 00:25:41.060 Logarithm in the in the vertical direction, ordinary 00:25:41.060 --> 00:25:42.970 in the x direction. 00:25:42.970 --> 00:25:44.720 OK. 00:25:44.720 --> 00:25:45.950 Good. 00:25:45.950 --> 00:25:52.380 Now I just want to add one sort of example. 00:25:52.380 --> 00:25:56.083 Because it's quite important and also quite practical. 00:25:56.083 --> 00:25:57.333 May I tell you about-- 00:26:02.840 --> 00:26:06.520 Let me ask you the question, and see if you get an idea. 00:26:06.520 --> 00:26:09.150 Because this is like basic to calculus. 00:26:09.150 --> 00:26:12.880 Let me talk about-- 00:26:12.880 --> 00:26:15.860 this e will stand for error. 00:26:15.860 --> 00:26:19.075 Error e. 00:26:19.075 --> 00:26:21.760 And what error am I talking about? 00:26:21.760 --> 00:26:26.120 I'm talking about the error as the difference between the 00:26:26.120 --> 00:26:28.490 derivative-- 00:26:28.490 --> 00:26:33.920 I have some function f of x. 00:26:33.920 --> 00:26:36.000 And there's its derivative. 00:26:36.000 --> 00:26:44.420 And I compare that with delta f over delta x. 00:26:44.420 --> 00:26:47.230 So what do I know? 00:26:47.230 --> 00:26:52.240 I know that as this is a function of delta x, I'm 00:26:52.240 --> 00:26:58.410 comparing the instant slope versus the average slope over 00:26:58.410 --> 00:27:02.040 a distance delta x. 00:27:02.040 --> 00:27:05.680 So it's not 0, right? 00:27:05.680 --> 00:27:10.040 This one is a finite movement. 00:27:10.040 --> 00:27:15.050 Delta x produces a finite moment delta f. 00:27:15.050 --> 00:27:20.180 As delta x goes to 0, that does approach this. 00:27:20.180 --> 00:27:22.720 So here's my question. 00:27:22.720 --> 00:27:27.990 My question is, this is approximately some constant 00:27:27.990 --> 00:27:32.680 times delta x to some power. 00:27:32.680 --> 00:27:34.815 And my question is, what is n? 00:27:40.910 --> 00:27:41.940 How close? 00:27:41.940 --> 00:27:47.340 What's a rough estimate of how near the delta f over delta x 00:27:47.340 --> 00:27:49.860 is to the actual derivative? 00:27:49.860 --> 00:27:51.180 OK. 00:27:51.180 --> 00:27:56.240 So I have to tell you what I meant by delta f over delta x. 00:27:56.240 --> 00:28:05.040 I meant what you also meant, f at x plus delta x minus f at x 00:28:05.040 --> 00:28:07.850 divided by delta x. 00:28:07.850 --> 00:28:10.720 In other words, that's the familiar delta f. 00:28:10.720 --> 00:28:14.470 Moving forward from x, I would call that a forward 00:28:14.470 --> 00:28:16.930 difference, a forward delta f. 00:28:16.930 --> 00:28:21.180 Because I'm starting at x, and I think of delta x as moving 00:28:21.180 --> 00:28:22.940 me a little bit forward. 00:28:22.940 --> 00:28:26.240 So I get the delta f, I divide by the delta x, and that's 00:28:26.240 --> 00:28:27.560 what this thing means. 00:28:27.560 --> 00:28:29.040 And do you know what n is? 00:28:33.620 --> 00:28:37.590 Let me connect it to my pictures. 00:28:37.590 --> 00:28:42.345 If I tried to graph this, I'd have a graph. 00:28:44.850 --> 00:28:45.690 You know. 00:28:45.690 --> 00:28:49.720 Here's my delta x and here's my e. 00:28:49.720 --> 00:28:53.750 This difference says delta x goes to 0, it goes to 0. 00:28:53.750 --> 00:28:57.370 You know, if delta x is small, e is small. 00:28:57.370 --> 00:29:00.390 If I divide delta x by 10, e divides by something. 00:29:04.060 --> 00:29:06.320 I don't even know if you see it on the camera. 00:29:06.320 --> 00:29:08.800 The graph has gone into a-- 00:29:08.800 --> 00:29:11.210 well, a black hole, or a chalk hole, or a 00:29:11.210 --> 00:29:12.740 white hole, or something. 00:29:12.740 --> 00:29:16.550 It's just completely invisible. 00:29:16.550 --> 00:29:18.940 I can't see the slope of this thing. 00:29:21.570 --> 00:29:26.720 But if I did it on log log paper, I'd see it clearly. 00:29:26.720 --> 00:29:30.420 And the answer would be 1. 00:29:30.420 --> 00:29:37.100 The error, the difference between derivative and average 00:29:37.100 --> 00:29:43.210 slope, goes like delta x to the first power. 00:29:43.210 --> 00:29:47.680 And then we can see later where that 1 comes from, and 00:29:47.680 --> 00:29:49.800 we can see where that a is. 00:29:49.800 --> 00:29:52.690 It's all in Taylor series. 00:29:52.690 --> 00:29:57.150 But here's my practical point. 00:29:57.150 --> 00:30:00.820 There is a much better delta f than this one. 00:30:00.820 --> 00:30:04.290 A much better delta f over delta x. 00:30:04.290 --> 00:30:09.090 An average slope that's much more accurate, and that in 00:30:09.090 --> 00:30:11.730 calculation I would always use. 00:30:11.730 --> 00:30:14.900 And the trouble with this one is, it's lopsided. 00:30:14.900 --> 00:30:15.920 It's one-sided. 00:30:15.920 --> 00:30:18.160 I only went forward. 00:30:18.160 --> 00:30:22.200 Or if delta x is negative, I'm only going backwards. 00:30:22.200 --> 00:30:25.920 And it turns out that the average of forward and 00:30:25.920 --> 00:30:31.200 backward is like centered at difference. 00:30:31.200 --> 00:30:34.070 So let me tell you a center difference. f at 00:30:34.070 --> 00:30:36.230 x plus delta x. 00:30:36.230 --> 00:30:40.690 So look a little forward, but take the difference from 00:30:40.690 --> 00:30:41.940 looking a little backward. 00:30:44.410 --> 00:30:48.380 That would be my change in f. 00:30:48.380 --> 00:30:54.720 But now what do I divide by to get a reasonable slope? 00:30:54.720 --> 00:31:00.050 Well, this is the change in f going from minus delta x-- 00:31:00.050 --> 00:31:03.370 delta x to the left of the point to delta x to the right 00:31:03.370 --> 00:31:04.350 of the point. 00:31:04.350 --> 00:31:09.840 The real movement there in the x-axis was a movement 00:31:09.840 --> 00:31:11.090 of two delta xs. 00:31:14.580 --> 00:31:17.200 So I would call this a center difference. 00:31:17.200 --> 00:31:19.099 Can I write that word "centered" down? 00:31:24.730 --> 00:31:29.810 And if I use that, which is a lot smarter if I'm practically 00:31:29.810 --> 00:31:33.050 wanting to get pictures, then what happens? 00:31:33.050 --> 00:31:37.350 So if this is now instead of this, instead of choosing this 00:31:37.350 --> 00:31:45.200 lopsided, simple, familiar but not that great difference, if 00:31:45.200 --> 00:31:53.710 I go for this one, the answer is, n changes to 2. 00:31:53.710 --> 00:31:56.350 n is 2 for this one. 00:31:56.350 --> 00:32:03.960 The accuracy is way, way better for center differences. 00:32:03.960 --> 00:32:09.510 And the point about the log graphs is, if I plot those 00:32:09.510 --> 00:32:14.170 points on the graph I would see that slope of 2 in the log 00:32:14.170 --> 00:32:17.460 log graph, it would be again-- 00:32:17.460 --> 00:32:22.780 in ordinary graph, it would become invisible 00:32:22.780 --> 00:32:24.690 as delta x got small. 00:32:24.690 --> 00:32:30.120 But on a log scale, I'd see it perfectly. 00:32:30.120 --> 00:32:31.300 OK. 00:32:31.300 --> 00:32:35.650 Some practical uses of logarithms. Now that we no 00:32:35.650 --> 00:32:39.550 longer use slide rules, this is what we do. 00:32:39.550 --> 00:32:41.650 Thanks. 00:32:41.650 --> 00:32:43.410 NARRATOR: This has been a production of MIT 00:32:43.410 --> 00:32:45.800 OpenCourseWare and Gilbert Strang. 00:32:45.800 --> 00:32:48.080 Funding for this video was provided by the Lord 00:32:48.080 --> 00:32:49.290 Foundation. 00:32:49.290 --> 00:32:52.420 To help OCW continue to provide free and open access 00:32:52.420 --> 00:32:55.500 to MIT courses, please make a donation at 00:32:55.500 --> 00:32:57.060 ocw.mit.edu/donate.