[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.12,0:00:01.59,Default,,0000,0000,0000,,- [Voiceover] We know\Nfrom previous videos,
Dialogue: 0,0:00:01.59,0:00:04.58,Default,,0000,0000,0000,,that the derivative with respect to X
Dialogue: 0,0:00:04.58,0:00:08.41,Default,,0000,0000,0000,,of the natural log of\NX, is equal to 1 over X.
Dialogue: 0,0:00:12.35,0:00:14.72,Default,,0000,0000,0000,,What I want to do in this\Nvideo is use that knowledge
Dialogue: 0,0:00:14.72,0:00:17.03,Default,,0000,0000,0000,,that we've seen in other\Nvideos to figure out
Dialogue: 0,0:00:17.03,0:00:19.93,Default,,0000,0000,0000,,what the derivative with respect to X is
Dialogue: 0,0:00:19.93,0:00:23.41,Default,,0000,0000,0000,,of a logarithm of an arbitrary base.
Dialogue: 0,0:00:23.41,0:00:27.16,Default,,0000,0000,0000,,So I'm just gonna call\Nthat log, base A of X.
Dialogue: 0,0:00:30.52,0:00:32.33,Default,,0000,0000,0000,,So how do we figure this out?
Dialogue: 0,0:00:32.33,0:00:35.20,Default,,0000,0000,0000,,Well, the key thing is, is\Nwhat you might be familiar with
Dialogue: 0,0:00:35.20,0:00:37.57,Default,,0000,0000,0000,,from your algebra or your\Npre calculus classes,
Dialogue: 0,0:00:37.57,0:00:40.19,Default,,0000,0000,0000,,which is having a change of base.
Dialogue: 0,0:00:40.19,0:00:43.31,Default,,0000,0000,0000,,So if I have some, I'll do it over here,
Dialogue: 0,0:00:43.31,0:00:47.60,Default,,0000,0000,0000,,log, base A of B, and\NI wanted to change it
Dialogue: 0,0:00:47.60,0:00:49.44,Default,,0000,0000,0000,,to a different base, let's\Nsay I wanna change it
Dialogue: 0,0:00:49.44,0:00:53.60,Default,,0000,0000,0000,,to base C, this is the same\Nthing as log, base C of B
Dialogue: 0,0:00:56.43,0:00:58.76,Default,,0000,0000,0000,,divided by log, base C of A.
Dialogue: 0,0:01:04.06,0:01:08.05,Default,,0000,0000,0000,,Log, base C of B, divided\Nby log, base C of A.
Dialogue: 0,0:01:08.05,0:01:09.78,Default,,0000,0000,0000,,This is a really useful\Nthing if you've never
Dialogue: 0,0:01:09.78,0:01:11.70,Default,,0000,0000,0000,,seen it before, you now have just seen it,
Dialogue: 0,0:01:11.70,0:01:13.87,Default,,0000,0000,0000,,this change of base, and we prove it
Dialogue: 0,0:01:13.87,0:01:16.52,Default,,0000,0000,0000,,in other videos on Khan Academy.
Dialogue: 0,0:01:16.52,0:01:18.50,Default,,0000,0000,0000,,But it's really useful\Nbecause, for example,
Dialogue: 0,0:01:18.50,0:01:21.01,Default,,0000,0000,0000,,your calculator has a log button.
Dialogue: 0,0:01:21.01,0:01:24.74,Default,,0000,0000,0000,,The log on your calculator\Nis log, base 10.
Dialogue: 0,0:01:24.74,0:01:28.56,Default,,0000,0000,0000,,So if you press 100 into your calculator
Dialogue: 0,0:01:28.56,0:01:31.73,Default,,0000,0000,0000,,and press log, you will get a 2 there.
Dialogue: 0,0:01:31.73,0:01:33.94,Default,,0000,0000,0000,,So whenever you just see log of 100,
Dialogue: 0,0:01:33.94,0:01:37.20,Default,,0000,0000,0000,,it's implicitly base 10,\Nand you also have a button
Dialogue: 0,0:01:37.20,0:01:40.37,Default,,0000,0000,0000,,for natural log, which is log, base E.
Dialogue: 0,0:01:40.37,0:01:44.20,Default,,0000,0000,0000,,Natural log of X is equal\Nto log, base E of X.
Dialogue: 0,0:01:46.03,0:01:48.15,Default,,0000,0000,0000,,But sometimes, you wanna\Nfind all sorts of different
Dialogue: 0,0:01:48.15,0:01:52.30,Default,,0000,0000,0000,,base logarithms and this is how you do it.
Dialogue: 0,0:01:52.30,0:01:54.72,Default,,0000,0000,0000,,So if you're using your\Ncalculator and you wanted to find
Dialogue: 0,0:01:54.72,0:01:58.06,Default,,0000,0000,0000,,what log, base 3 of 8 is, you would say,
Dialogue: 0,0:02:01.49,0:02:05.98,Default,,0000,0000,0000,,you would type in your\Ncalculator log of 8 and log of 3.
Dialogue: 0,0:02:05.98,0:02:09.25,Default,,0000,0000,0000,,Or, let me write it this way, and log of 3
Dialogue: 0,0:02:09.25,0:02:11.70,Default,,0000,0000,0000,,where both of these\Nare implicitly base 10,
Dialogue: 0,0:02:11.70,0:02:13.12,Default,,0000,0000,0000,,and you'd get the same value if you did
Dialogue: 0,0:02:13.12,0:02:16.87,Default,,0000,0000,0000,,natural log of 8 divided\Nby natural log of 3.
Dialogue: 0,0:02:17.94,0:02:19.92,Default,,0000,0000,0000,,Which you might also\Nhave on your calculator.
Dialogue: 0,0:02:19.92,0:02:23.99,Default,,0000,0000,0000,,And what we're gonna do\Nin this video is leverage
Dialogue: 0,0:02:23.99,0:02:26.41,Default,,0000,0000,0000,,the natural log because we\Nknow what the derivative
Dialogue: 0,0:02:26.41,0:02:28.59,Default,,0000,0000,0000,,of the natural log is.
Dialogue: 0,0:02:28.59,0:02:31.93,Default,,0000,0000,0000,,So this derivative is the\Nsame thing as the derivative
Dialogue: 0,0:02:31.93,0:02:33.68,Default,,0000,0000,0000,,with respect to X of.
Dialogue: 0,0:02:35.27,0:02:38.39,Default,,0000,0000,0000,,Well log, base A of X, can be rewritten as
Dialogue: 0,0:02:38.39,0:02:41.64,Default,,0000,0000,0000,,natural log of X over natural log of A.
Dialogue: 0,0:02:43.99,0:02:46.06,Default,,0000,0000,0000,,And now natural log of\NA, that's just a number.
Dialogue: 0,0:02:46.06,0:02:50.63,Default,,0000,0000,0000,,I could rewrite this as,\Nlet me write it this way.
Dialogue: 0,0:02:50.63,0:02:54.71,Default,,0000,0000,0000,,One over natural log of\NA times natural log of X.
Dialogue: 0,0:02:56.45,0:02:58.09,Default,,0000,0000,0000,,And what's the derivative of that?
Dialogue: 0,0:02:58.09,0:03:00.13,Default,,0000,0000,0000,,We could just take the constant out.
Dialogue: 0,0:03:00.13,0:03:02.36,Default,,0000,0000,0000,,One over natural log of\NA, that's just a number.
Dialogue: 0,0:03:02.36,0:03:06.44,Default,,0000,0000,0000,,So we're gonna get 1\Nover the natural log of A
Dialogue: 0,0:03:06.44,0:03:10.60,Default,,0000,0000,0000,,times the derivative with\Nrespect to X of natural log of X.
Dialogue: 0,0:03:15.02,0:03:16.68,Default,,0000,0000,0000,,Of natural log of X.
Dialogue: 0,0:03:18.09,0:03:20.26,Default,,0000,0000,0000,,Which we already know is 1 over X.
Dialogue: 0,0:03:20.26,0:03:23.13,Default,,0000,0000,0000,,So this thing right\Nover here, is 1 over X.
Dialogue: 0,0:03:23.13,0:03:27.30,Default,,0000,0000,0000,,So what we get is 1 over\Nnatural log of A times 1 over X.
Dialogue: 0,0:03:29.24,0:03:33.40,Default,,0000,0000,0000,,Which we could write as, 1\Nover natural log of A times X.
Dialogue: 0,0:03:40.41,0:03:42.39,Default,,0000,0000,0000,,Which is a really useful thing to know.
Dialogue: 0,0:03:42.39,0:03:46.32,Default,,0000,0000,0000,,So now, we could take\Nall sorts of derivatives.
Dialogue: 0,0:03:46.32,0:03:50.48,Default,,0000,0000,0000,,So if I were to tell you F of\NX is equal to log, base 7 of X
Dialogue: 0,0:03:56.65,0:04:01.08,Default,,0000,0000,0000,,well now we can say well F\Nprime of X is going to be
Dialogue: 0,0:04:01.08,0:04:04.08,Default,,0000,0000,0000,,1 over the natural log of 7 times X.
Dialogue: 0,0:04:07.05,0:04:11.20,Default,,0000,0000,0000,,If we had a constant out\Nfront, if we had for example,
Dialogue: 0,0:04:11.20,0:04:12.65,Default,,0000,0000,0000,,G of X.
Dialogue: 0,0:04:12.65,0:04:16.82,Default,,0000,0000,0000,,G of X is equal to negative\N3 times log, base, I know.
Dialogue: 0,0:04:18.50,0:04:20.39,Default,,0000,0000,0000,,Log, base pi.
Dialogue: 0,0:04:20.39,0:04:21.56,Default,,0000,0000,0000,,Pi is a number.
Dialogue: 0,0:04:21.56,0:04:25.73,Default,,0000,0000,0000,,Log, base pi of X, well G\Nprime of X would be equal to
Dialogue: 0,0:04:28.34,0:04:29.25,Default,,0000,0000,0000,,1 over, oh.
Dialogue: 0,0:04:30.51,0:04:32.01,Default,,0000,0000,0000,,Let me be careful, I have\Nthis constant out here.
Dialogue: 0,0:04:32.01,0:04:35.91,Default,,0000,0000,0000,,So it'd be negative 3 over,\Nit's just that negative 3,
Dialogue: 0,0:04:35.91,0:04:38.16,Default,,0000,0000,0000,,over the natural log of pi.
Dialogue: 0,0:04:40.26,0:04:42.13,Default,,0000,0000,0000,,This is the natural log of this number.
Dialogue: 0,0:04:42.13,0:04:42.96,Default,,0000,0000,0000,,Times X.
Dialogue: 0,0:04:44.19,0:04:46.88,Default,,0000,0000,0000,,So hopefully, that gives\Nyou a hang of things.