1 00:00:00,117 --> 00:00:01,593 - [Voiceover] We know from previous videos, 2 00:00:01,593 --> 00:00:04,575 that the derivative with respect to X 3 00:00:04,575 --> 00:00:08,408 of the natural log of X, is equal to 1 over X. 4 00:00:12,350 --> 00:00:14,718 What I want to do in this video is use that knowledge 5 00:00:14,718 --> 00:00:17,031 that we've seen in other videos to figure out 6 00:00:17,031 --> 00:00:19,929 what the derivative with respect to X is 7 00:00:19,929 --> 00:00:23,412 of a logarithm of an arbitrary base. 8 00:00:23,412 --> 00:00:27,162 So I'm just gonna call that log, base A of X. 9 00:00:30,517 --> 00:00:32,328 So how do we figure this out? 10 00:00:32,328 --> 00:00:35,198 Well, the key thing is, is what you might be familiar with 11 00:00:35,198 --> 00:00:37,566 from your algebra or your pre calculus classes, 12 00:00:37,566 --> 00:00:40,186 which is having a change of base. 13 00:00:40,186 --> 00:00:43,306 So if I have some, I'll do it over here, 14 00:00:43,306 --> 00:00:47,597 log, base A of B, and I wanted to change it 15 00:00:47,597 --> 00:00:49,436 to a different base, let's say I wanna change it 16 00:00:49,436 --> 00:00:53,603 to base C, this is the same thing as log, base C of B 17 00:00:56,430 --> 00:00:58,763 divided by log, base C of A. 18 00:01:04,065 --> 00:01:08,050 Log, base C of B, divided by log, base C of A. 19 00:01:08,050 --> 00:01:09,777 This is a really useful thing if you've never 20 00:01:09,777 --> 00:01:11,700 seen it before, you now have just seen it, 21 00:01:11,700 --> 00:01:13,873 this change of base, and we prove it 22 00:01:13,873 --> 00:01:16,520 in other videos on Khan Academy. 23 00:01:16,520 --> 00:01:18,499 But it's really useful because, for example, 24 00:01:18,499 --> 00:01:21,006 your calculator has a log button. 25 00:01:21,006 --> 00:01:24,740 The log on your calculator is log, base 10. 26 00:01:24,740 --> 00:01:28,557 So if you press 100 into your calculator 27 00:01:28,557 --> 00:01:31,734 and press log, you will get a 2 there. 28 00:01:31,734 --> 00:01:33,935 So whenever you just see log of 100, 29 00:01:33,935 --> 00:01:37,195 it's implicitly base 10, and you also have a button 30 00:01:37,195 --> 00:01:40,372 for natural log, which is log, base E. 31 00:01:40,372 --> 00:01:44,205 Natural log of X is equal to log, base E of X. 32 00:01:46,028 --> 00:01:48,146 But sometimes, you wanna find all sorts of different 33 00:01:48,146 --> 00:01:52,298 base logarithms and this is how you do it. 34 00:01:52,298 --> 00:01:54,722 So if you're using your calculator and you wanted to find 35 00:01:54,722 --> 00:01:58,055 what log, base 3 of 8 is, you would say, 36 00:02:01,493 --> 00:02:05,979 you would type in your calculator log of 8 and log of 3. 37 00:02:05,979 --> 00:02:09,247 Or, let me write it this way, and log of 3 38 00:02:09,247 --> 00:02:11,700 where both of these are implicitly base 10, 39 00:02:11,700 --> 00:02:13,121 and you'd get the same value if you did 40 00:02:13,121 --> 00:02:16,871 natural log of 8 divided by natural log of 3. 41 00:02:17,942 --> 00:02:19,920 Which you might also have on your calculator. 42 00:02:19,920 --> 00:02:23,988 And what we're gonna do in this video is leverage 43 00:02:23,988 --> 00:02:26,412 the natural log because we know what the derivative 44 00:02:26,412 --> 00:02:28,586 of the natural log is. 45 00:02:28,586 --> 00:02:31,929 So this derivative is the same thing as the derivative 46 00:02:31,929 --> 00:02:33,679 with respect to X of. 47 00:02:35,273 --> 00:02:38,394 Well log, base A of X, can be rewritten as 48 00:02:38,394 --> 00:02:41,644 natural log of X over natural log of A. 49 00:02:43,994 --> 00:02:46,056 And now natural log of A, that's just a number. 50 00:02:46,056 --> 00:02:50,626 I could rewrite this as, let me write it this way. 51 00:02:50,626 --> 00:02:54,709 One over natural log of A times natural log of X. 52 00:02:56,450 --> 00:02:58,094 And what's the derivative of that? 53 00:02:58,094 --> 00:03:00,128 We could just take the constant out. 54 00:03:00,128 --> 00:03:02,357 One over natural log of A, that's just a number. 55 00:03:02,357 --> 00:03:06,435 So we're gonna get 1 over the natural log of A 56 00:03:06,435 --> 00:03:10,602 times the derivative with respect to X of natural log of X. 57 00:03:15,017 --> 00:03:16,684 Of natural log of X. 58 00:03:18,090 --> 00:03:20,264 Which we already know is 1 over X. 59 00:03:20,264 --> 00:03:23,134 So this thing right over here, is 1 over X. 60 00:03:23,134 --> 00:03:27,301 So what we get is 1 over natural log of A times 1 over X. 61 00:03:29,236 --> 00:03:33,403 Which we could write as, 1 over natural log of A times X. 62 00:03:40,409 --> 00:03:42,388 Which is a really useful thing to know. 63 00:03:42,388 --> 00:03:46,317 So now, we could take all sorts of derivatives. 64 00:03:46,317 --> 00:03:50,484 So if I were to tell you F of X is equal to log, base 7 of X 65 00:03:56,654 --> 00:04:01,084 well now we can say well F prime of X is going to be 66 00:04:01,084 --> 00:04:04,084 1 over the natural log of 7 times X. 67 00:04:07,047 --> 00:04:11,199 If we had a constant out front, if we had for example, 68 00:04:11,199 --> 00:04:12,648 G of X. 69 00:04:12,648 --> 00:04:16,815 G of X is equal to negative 3 times log, base, I know. 70 00:04:18,499 --> 00:04:20,394 Log, base pi. 71 00:04:20,394 --> 00:04:21,564 Pi is a number. 72 00:04:21,564 --> 00:04:25,731 Log, base pi of X, well G prime of X would be equal to 73 00:04:28,335 --> 00:04:29,252 1 over, oh. 74 00:04:30,509 --> 00:04:32,013 Let me be careful, I have this constant out here. 75 00:04:32,013 --> 00:04:35,914 So it'd be negative 3 over, it's just that negative 3, 76 00:04:35,914 --> 00:04:38,164 over the natural log of pi. 77 00:04:40,261 --> 00:04:42,128 This is the natural log of this number. 78 00:04:42,128 --> 00:04:42,961 Times X. 79 00:04:44,190 --> 00:04:46,879 So hopefully, that gives you a hang of things.