[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:02.40,0:00:04.88,Default,,0000,0000,0000,,[INSTRUCTOR] All\Nright, logarithms.
Dialogue: 0,0:00:05.52,0:00:09.92,Default,,0000,0000,0000,,Logarithms, why we need\Nthem is exactly this right
Dialogue: 0,0:00:10.08,0:00:11.44,Default,,0000,0000,0000,,here, what we're gonna see.
Dialogue: 0,0:00:11.68,0:00:13.20,Default,,0000,0000,0000,,If we're trying to solve for x
Dialogue: 0,0:00:13.60,0:00:17.60,Default,,0000,0000,0000,,And we have 2 to some\Npower is 16, I can think,
Dialogue: 0,0:00:17.60,0:00:19.52,Default,,0000,0000,0000,,okay, well, 2 times what?
Dialogue: 0,0:00:19.52,0:00:21.44,Default,,0000,0000,0000,,2 times itself how many times?
Dialogue: 0,0:00:21.60,0:00:24.88,Default,,0000,0000,0000,,Well, 2 times itself\Nfour times is 16,
Dialogue: 0,0:00:24.88,0:00:27.44,Default,,0000,0000,0000,,and so x has gotta\Nbe equal to 4.
Dialogue: 0,0:00:28.40,0:00:31.52,Default,,0000,0000,0000,,That one's not bad when\Nit works out equally.
Dialogue: 0,0:00:31.68,0:00:34.80,Default,,0000,0000,0000,,However, 2 to what power is 20?
Dialogue: 0,0:00:35.12,0:00:38.96,Default,,0000,0000,0000,,Well, we just said 2\Nto the fourth was 16,
Dialogue: 0,0:00:39.28,0:00:43.68,Default,,0000,0000,0000,,and we said we know that 2\Nto the fifth then is times
Dialogue: 0,0:00:43.76,0:00:46.08,Default,,0000,0000,0000,,another 2, which is 32.
Dialogue: 0,0:00:46.24,0:00:49.12,Default,,0000,0000,0000,,And so 20 falls someplace\Nin between there.
Dialogue: 0,0:00:49.36,0:00:56.24,Default,,0000,0000,0000,,And so we know that x is\Nsomeplace in between 4 and 5.
Dialogue: 0,0:00:56.64,0:00:59.68,Default,,0000,0000,0000,,But how exact can we get\Nwithout saying, okay, well,
Dialogue: 0,0:00:59.68,0:01:04.48,Default,,0000,0000,0000,,2 to the 4.2, 2 to the 4.3,\N4.4, and doing that,
Dialogue: 0,0:01:04.64,0:01:06.24,Default,,0000,0000,0000,,we need logarithms.
Dialogue: 0,0:01:06.64,0:01:08.32,Default,,0000,0000,0000,,Same deal with this one.
Dialogue: 0,0:01:08.88,0:01:11.52,Default,,0000,0000,0000,,We know that 2 to the fifth\Npower, like we just said,
Dialogue: 0,0:01:11.52,0:01:16.80,Default,,0000,0000,0000,,was 32, and we know that 2\Nto the sixth power is times
Dialogue: 0,0:01:16.88,0:01:19.04,Default,,0000,0000,0000,,another 2, which is 64.
Dialogue: 0,0:01:19.28,0:01:25.28,Default,,0000,0000,0000,,And so we know that 2 to\Nthe 5 point something,
Dialogue: 0,0:01:25.52,0:01:30.48,Default,,0000,0000,0000,,that x is in between\N5 and 6 for this one.
Dialogue: 0,0:01:31.04,0:01:34.08,Default,,0000,0000,0000,,But what exactly we\Nneed logarithms for?
Dialogue: 0,0:01:34.32,0:01:39.04,Default,,0000,0000,0000,,So, let's explore the logarithm\Nbutton a little bit here.
Dialogue: 0,0:01:39.84,0:01:43.36,Default,,0000,0000,0000,,So 10 to the 0,\Nanything to the 0 power
Dialogue: 0,0:01:43.68,0:01:48.16,Default,,0000,0000,0000,,is 1. 10 to the first,\N10 times itself.
Dialogue: 0,0:01:48.32,0:01:51.20,Default,,0000,0000,0000,,That 10 squared, 100.
Dialogue: 0,0:01:51.28,0:01:56.48,Default,,0000,0000,0000,,10 cubed, 10 times 10 times 10,\Nso we've got 1,000.
Dialogue: 0,0:01:56.64,0:02:02.48,Default,,0000,0000,0000,,And 10 to the fourth is\N1, 2, 3, 4, we can just
Dialogue: 0,0:02:02.64,0:02:04.72,Default,,0000,0000,0000,,keep adding those zeros.
Dialogue: 0,0:02:05.20,0:02:12.24,Default,,0000,0000,0000,,So, we've got this log button,\Nfancy log button down here.
Dialogue: 0,0:02:13.20,0:02:15.28,Default,,0000,0000,0000,,And so let's take the log of 1.
Dialogue: 0,0:02:15.60,0:02:17.36,Default,,0000,0000,0000,,Log of 1 is 0.
Dialogue: 0,0:02:17.68,0:02:19.28,Default,,0000,0000,0000,,Log of 10.
Dialogue: 0,0:02:20.24,0:02:22.64,Default,,0000,0000,0000,,Log of 100.
Dialogue: 0,0:02:24.32,0:02:26.88,Default,,0000,0000,0000,,Log of 1,000.
Dialogue: 0,0:02:27.04,0:02:29.04,Default,,0000,0000,0000,,And you can almost\Nguess what the log of
Dialogue: 0,0:02:29.20,0:02:32.32,Default,,0000,0000,0000,,10,000 is going to be.
Dialogue: 0,0:02:32.64,0:02:38.96,Default,,0000,0000,0000,,So log of 1 is 0, 1, 2, 3, 4.
Dialogue: 0,0:02:39.20,0:02:41.36,Default,,0000,0000,0000,,So what do you think\Nthe log button does?
Dialogue: 0,0:02:41.44,0:02:44.32,Default,,0000,0000,0000,,Well, what I want us to draw\Nour attention to is this
Dialogue: 0,0:02:44.48,0:02:48.48,Default,,0000,0000,0000,,0 is the same as this 0.
Dialogue: 0,0:02:48.72,0:02:58.80,Default,,0000,0000,0000,,This one, that one,\N2 and 2, 3 and 3, 4, and 4.
Dialogue: 0,0:02:59.92,0:03:02.32,Default,,0000,0000,0000,,So, what do we think\Nthe log button does?
Dialogue: 0,0:03:02.32,0:03:07.52,Default,,0000,0000,0000,,Well, if we take 10\Nto the 0 we get 1,
Dialogue: 0,0:03:07.52,0:03:09.76,Default,,0000,0000,0000,,and if we take the\Nlog of 1 it gives us
Dialogue: 0,0:03:09.84,0:03:11.28,Default,,0000,0000,0000,,that exponent again.
Dialogue: 0,0:03:11.60,0:03:19.60,Default,,0000,0000,0000,,And so the log is\Nundoing, it undoes
Dialogue: 0,0:03:23.44,0:03:27.68,Default,,0000,0000,0000,,the exponent.
Dialogue: 0,0:03:28.56,0:03:31.28,Default,,0000,0000,0000,,They are inverse functions
Dialogue: 0,0:03:41.60,0:03:50.00,Default,,0000,0000,0000,,of exponents, or exponentials.
Dialogue: 0,0:03:50.88,0:03:52.96,Default,,0000,0000,0000,,So here's what I mean.
Dialogue: 0,0:03:53.44,0:03:54.16,Default,,0000,0000,0000,,If you have
Dialogue: 0,0:04:00.24,0:04:05.84,Default,,0000,0000,0000,,b to the x equals a,\Nwe can write that,
Dialogue: 0,0:04:05.84,0:04:09.12,Default,,0000,0000,0000,,and this is called\Nexponential form.
Dialogue: 0,0:04:11.76,0:04:15.60,Default,,0000,0000,0000,,We can write that as log\Nform, saying the log,
Dialogue: 0,0:04:15.84,0:04:18.24,Default,,0000,0000,0000,,which is just another operation.
Dialogue: 0,0:04:18.88,0:04:19.60,Default,,0000,0000,0000,,Base b
Dialogue: 0,0:04:22.08,0:04:26.16,Default,,0000,0000,0000,,of a equals x.
Dialogue: 0,0:04:26.40,0:04:32.56,Default,,0000,0000,0000,,And so this b, we call\Nthe base of the log.
Dialogue: 0,0:04:33.04,0:04:35.84,Default,,0000,0000,0000,,And this is what we're\Ntaking the log of f.
Dialogue: 0,0:04:36.32,0:04:41.04,Default,,0000,0000,0000,,And so the base of the log\Nand the base of the exponent
Dialogue: 0,0:04:42.16,0:04:46.40,Default,,0000,0000,0000,,are the exact same thing, and\Nthen the x and the a swap sides.
Dialogue: 0,0:04:48.16,0:04:52.80,Default,,0000,0000,0000,,If you guys can know this and go\Nback and forth from this form,
Dialogue: 0,0:04:52.96,0:04:57.60,Default,,0000,0000,0000,,you're going to go extremely\Nfar with this logarithm concept.
Dialogue: 0,0:04:58.16,0:05:01.04,Default,,0000,0000,0000,,To know that they\Nare the opposite of
Dialogue: 0,0:05:01.12,0:05:02.40,Default,,0000,0000,0000,,each other like that.
Dialogue: 0,0:05:02.88,0:05:04.56,Default,,0000,0000,0000,,And so what we're gonna\Ndo is just practice
Dialogue: 0,0:05:04.64,0:05:06.08,Default,,0000,0000,0000,,rewriting this like this.
Dialogue: 0,0:05:06.40,0:05:09.52,Default,,0000,0000,0000,,So, rewrite these as logs.
Dialogue: 0,0:05:09.84,0:05:11.92,Default,,0000,0000,0000,,10 to the third is a 1,000.
Dialogue: 0,0:05:12.00,0:05:17.12,Default,,0000,0000,0000,,So, the log base 10,\Nbecause the base of the
Dialogue: 0,0:05:17.20,0:05:24.96,Default,,0000,0000,0000,,exponent becomes the base of\Nthe log, of 1,000 equals 3.
Dialogue: 0,0:05:25.68,0:05:29.36,Default,,0000,0000,0000,,The log base 5,\Nthe base of the
Dialogue: 0,0:05:29.52,0:05:34.40,Default,,0000,0000,0000,,exponent of 625 equals 4.
Dialogue: 0,0:05:35.76,0:05:44.64,Default,,0000,0000,0000,,Log base 2 of 1,024 equals 10.
Dialogue: 0,0:05:45.12,0:05:48.48,Default,,0000,0000,0000,,And so now we wanna find x,\Nand so this is what we were
Dialogue: 0,0:05:48.64,0:05:49.60,Default,,0000,0000,0000,,talking about before.
Dialogue: 0,0:05:50.08,0:05:59.20,Default,,0000,0000,0000,,Log base 2 of 16 equals x.\NBase of the exponent becomes
Dialogue: 0,0:05:59.20,0:06:02.32,Default,,0000,0000,0000,,the base of the log, and now\Nthe x is all by itself.
Dialogue: 0,0:06:02.40,0:06:06.00,Default,,0000,0000,0000,,So if only we could\Nevaluate that.
Dialogue: 0,0:06:06.16,0:06:08.96,Default,,0000,0000,0000,,Now, we didn't really\Nneed logs for that,
Dialogue: 0,0:06:09.20,0:06:11.52,Default,,0000,0000,0000,,and our calculator\Ncan't do that outright,
Dialogue: 0,0:06:11.60,0:06:14.16,Default,,0000,0000,0000,,but I'll show you how we\Ncan adjust it for it.
Dialogue: 0,0:06:15.36,0:06:22.24,Default,,0000,0000,0000,,Log base 2 of 20 equals\Nx, base of the log,
Dialogue: 0,0:06:22.40,0:06:28.48,Default,,0000,0000,0000,,base of the exponent,\Nlog base 2 of 50 equals x.
Dialogue: 0,0:06:28.56,0:06:32.72,Default,,0000,0000,0000,,Let's get really good at\Nchanging back and forth
Dialogue: 0,0:06:32.80,0:06:34.72,Default,,0000,0000,0000,,between those two things.
Dialogue: 0,0:06:35.68,0:06:39.84,Default,,0000,0000,0000,,So, something we need\Nto know, the common log.
Dialogue: 0,0:06:40.16,0:06:43.44,Default,,0000,0000,0000,,Call it the common log and\Nthat's what the log button
Dialogue: 0,0:06:43.44,0:06:45.60,Default,,0000,0000,0000,,on our calculator is,\Nbecause notice the log
Dialogue: 0,0:06:45.60,0:06:47.76,Default,,0000,0000,0000,,button on your calculator\Ndoesn't have a number or
Dialogue: 0,0:06:47.84,0:06:53.68,Default,,0000,0000,0000,,a base, it's because it\Nautomatically does log base 10.
Dialogue: 0,0:06:56.64,0:06:59.44,Default,,0000,0000,0000,,Because a ton of our numbers\Nare in the base 10 system,
Dialogue: 0,0:06:59.44,0:07:02.80,Default,,0000,0000,0000,,we work in the base 10 system,\Nand so that's why it's on there.
Dialogue: 0,0:07:03.20,0:07:06.72,Default,,0000,0000,0000,,And the natural log,\Nwe already talked about
Dialogue: 0,0:07:06.80,0:07:09.20,Default,,0000,0000,0000,,the natural number being e.
Dialogue: 0,0:07:09.36,0:07:14.80,Default,,0000,0000,0000,,And so the natural log is\Nany log that has the base e.
Dialogue: 0,0:07:15.36,0:07:19.12,Default,,0000,0000,0000,,And so, in both cases,\Nwe don't write the bases,
Dialogue: 0,0:07:19.20,0:07:21.84,Default,,0000,0000,0000,,and it's a little easier\Nto recognize, but you need
Dialogue: 0,0:07:21.92,0:07:24.80,Default,,0000,0000,0000,,to notice what the base is.
Dialogue: 0,0:07:25.44,0:07:29.36,Default,,0000,0000,0000,,So what if it doesn't\Nhave base 10 or base e,
Dialogue: 0,0:07:29.44,0:07:32.88,Default,,0000,0000,0000,,and we can't use the fancy\Nbuttons on the calculator,
Dialogue: 0,0:07:32.96,0:07:34.72,Default,,0000,0000,0000,,log and natural log?
Dialogue: 0,0:07:35.04,0:07:38.24,Default,,0000,0000,0000,,Well, we use the\Nchange of base formula.
Dialogue: 0,0:07:38.56,0:07:44.48,Default,,0000,0000,0000,,And so we can change any\Nbase, a, into base 10.
Dialogue: 0,0:07:45.12,0:07:52.16,Default,,0000,0000,0000,,And we do the log of x\Ndivided by the log of a.
Dialogue: 0,0:07:53.04,0:07:55.12,Default,,0000,0000,0000,,Or you could use the\Nnatural log if you wanted
Dialogue: 0,0:07:55.12,0:07:56.88,Default,,0000,0000,0000,,to, whichever your preference.
Dialogue: 0,0:07:56.96,0:07:58.72,Default,,0000,0000,0000,,They give you the\Nexact same answer.
Dialogue: 0,0:07:58.88,0:08:02.96,Default,,0000,0000,0000,,Natural log of x,\Nnatural log of a.
Dialogue: 0,0:08:03.04,0:08:04.88,Default,,0000,0000,0000,,And the whole reason\Nbehind it is because
Dialogue: 0,0:08:05.04,0:08:08.08,Default,,0000,0000,0000,,you can really change it\Nto any base you want to.
Dialogue: 0,0:08:08.64,0:08:12.96,Default,,0000,0000,0000,,Log of x divided by log\Nof a. This could be base
Dialogue: 0,0:08:13.04,0:08:15.60,Default,,0000,0000,0000,,b and base b, as long as\Nthey're the same base.
Dialogue: 0,0:08:15.76,0:08:19.44,Default,,0000,0000,0000,,But most often,\Nwe use log base b.
Dialogue: 0,0:08:19.52,0:08:20.88,Default,,0000,0000,0000,,So let's utilize this.
Dialogue: 0,0:08:20.96,0:08:22.88,Default,,0000,0000,0000,,Let's see what we can get.
Dialogue: 0,0:08:23.20,0:08:26.16,Default,,0000,0000,0000,,Let's go back to our\Nother page and say
Dialogue: 0,0:08:28.72,0:08:34.64,Default,,0000,0000,0000,,log base 16, log of\N16 divided by log of 2.
Dialogue: 0,0:08:35.28,0:08:36.56,Default,,0000,0000,0000,,And that's 4.
Dialogue: 0,0:08:36.80,0:08:38.80,Default,,0000,0000,0000,,And so that's what\Nwe got before.
Dialogue: 0,0:08:41.52,0:08:46.32,Default,,0000,0000,0000,,Log 20 divided by log 2.
Dialogue: 0,0:08:46.48,0:08:49.12,Default,,0000,0000,0000,,We said this one was\Nin between 4 and 5,
Dialogue: 0,0:08:49.28,0:08:54.08,Default,,0000,0000,0000,,and so that one's\Napproximately 4.322.
Dialogue: 0,0:08:54.40,0:08:59.28,Default,,0000,0000,0000,,So that if we go back to why\Nwe were saying that, 2 to the
Dialogue: 0,0:08:59.44,0:09:04.24,Default,,0000,0000,0000,,power of 4.322 is about 20.
Dialogue: 0,0:09:04.80,0:09:08.40,Default,,0000,0000,0000,,And then finally,\Nlog base 2 of 50.
Dialogue: 0,0:09:08.80,0:09:10.56,Default,,0000,0000,0000,,Let's do the natural\Nlog this time.
Dialogue: 0,0:09:10.56,0:09:15.04,Default,,0000,0000,0000,,Natural log of 50 divided\Nby the natural log of 2.
Dialogue: 0,0:09:15.44,0:09:16.64,Default,,0000,0000,0000,,And just to show you
Dialogue: 0,0:09:18.80,0:09:20.32,Default,,0000,0000,0000,,that we get the\Nexact same thing,
Dialogue: 0,0:09:20.32,0:09:22.08,Default,,0000,0000,0000,,no matter which\Nway we do it.
Dialogue: 0,0:09:22.80,0:09:27.12,Default,,0000,0000,0000,,5.644.
Dialogue: 0,0:09:27.68,0:09:31.44,Default,,0000,0000,0000,,All right, we'll come back\Nand talk more logarithms.