What is a Natural Log Ln(x)? - Part 1 (Logarithm w/ Base e - Euler's number) - Free Educational videos for Students in K-12

What is a Natural Log Ln(x)? - Part 1 (Logarithm w/ Base e - Euler's number) - By Math and Science

Transcript


00:00	Hello , welcome back . The title of this lesson
00:02	is called the natural logarithms part one . Now I'm
00:05	really excited to teach this lesson because what usually happens
00:08	in math is you go through algebra , you go
00:10	through trig and Pri Khao and calculus and right around
00:13	that time you start to bump into these logarithms all
00:16	the time . And then more specifically when you get
00:18	in the calculus you start bumping into the natural algorithm
00:21	all the time . I mean almost every problem ,
00:24	especially in advanced math ends up involving a natural log
00:27	or an exponential function with this base E that we've
00:30	been talking about uh over and over and over again
00:33	. So it just kind of comes out of nowhere
00:35	. Now as we discussed in the last lesson ,
00:37	the number E oilers Number E is a 2.71 in
00:41	an infinite decimals , infinite non repeating decimals . After
00:44	that it's a number that has equal stature with the
00:47	number pi it's just , we don't usually learn about
00:49	E until later on in math because you're usually not
00:52	learning exponential functions . In fifth grade fifth grade you
00:55	have to learn about pie because you have to learn
00:57	about circles and now that we're learning about exponential ,
00:59	you're never ever going to get away from this number
01:01	E . It's just that important . All right ,
01:04	so in this lesson , what we're gonna do ,
01:05	I'm gonna do a brief recap of where the number
01:07	comes from . We already covered that extensively in the
01:10	last lesson , but it's worth saying in a little
01:12	condensed version again what that is . And then we're
01:15	going to dive more specifically into the natural logarithms .
01:18	So , you know exactly what it looks like exactly
01:20	what the shape of that curve is , where it
01:22	comes from and how it relates to the exponential function
01:25	there . And then we'll solve a few problems dealing
01:27	with the algorithms that have this natural uh this natural
01:31	not this natural number . This uh this this irrational
01:34	number E as the base , the natural algorithm .
01:37	Now , the good news is all of the laws
01:39	of logarithms that we studied , all of them .
01:42	They all apply to natural algorithms . Never ever forget
01:45	that a natural algorithm is just a regular old law
01:48	algorithm . It's just that it has a very special
01:51	number E for that base . Now , let's do
01:53	a little recap . I'm gonna say recall because we
01:57	did cover this , I did an entire lesson on
01:59	where the number E comes from . There's this number
02:01	E in math , it's just as important as pie
02:04	and it's approximately equal to 2.71828 dot dot dot dot
02:10	dots . Mean that this is an irrational number .
02:13	I can compute e 2 50 million decimal places and
02:16	I will never get a repeating pattern in the decimals
02:19	. The decimal will go on forever , but they'll
02:21	never repeat just like pie . So that's why it's
02:23	irrational . Then as it's called , uh , the
02:27	number E comes from the fact that mathematician by the
02:29	name of Oiler calculated this number and we went through
02:33	the calculation in the last lesson , I'll kind of
02:35	recap it for you here . But basically the number
02:38	E comes from looking at how money grows exponentially .
02:41	So you start off with that exponential growth of money
02:44	and we look at compounding the interest rate more and
02:48	more and more often , eventually compounding every nanosecond compounding
02:52	continuously and out of those calculations fall out this and
02:55	this number eight . Let me take a little bit
02:58	of an aside . A lot of students will say
03:00	, well why is that number E 2.718 to 8
03:04	. And why is it that exact number ? That's
03:05	a weird number . I agree with you . It's
03:07	weird but you then you have to turn it around
03:09	and say , well why is pi the number that
03:12	it is 3.14159 and the infinite decimal places after that
03:15	? Why is pi equal to that ? Why is
03:18	pie not 3.17 ? And then a bunch of numbers
03:20	at some point in math , you have to just
03:23	accept that our universe has certain numbers and certain characteristics
03:26	that are just the way that they are in our
03:29	universe . The number pi comes about from looking at
03:32	the circumference of a circle and the diameter of the
03:35	circle . The that conversion ratio there in circumference equals
03:39	pi times diameter . That's what comes about there .
03:42	For the definition of pie . It has to do
03:44	with the geometry of our universe . That's why Pie
03:46	is the way that it is . Why is the
03:48	number Either way it is . It has to do
03:50	with the way things grow exponentially in our universe .
03:53	And the laws of physics and mathematics of our universe
03:55	governed things growing exponentially . And this number , it
03:58	is a very special number that yields a very special
04:02	exponential growth curve with very special characteristics . Why is
04:05	it exactly that number ? Nobody knows . But it's
04:07	the way it is in our universe . And so
04:09	we have to kind of just accept that and start
04:10	to use it going forward . You're never gonna get
04:12	away from the number E . It's just too important
04:16	. All right . So this number E is important
04:18	because of a few things because once I know what
04:22	this number is , then I can define a function
04:26	. This is an exponential function E . To the
04:28	X . Remember we can define exponential functions with any
04:31	base I want , I can make a function F
04:33	of X is equal to 10 to the power of
04:35	X . Or f of X is equal to five
04:37	to the power of X . Or f of X
04:38	is equal to 37 to the power of X .
04:40	When I say F of X is E to the
04:42	X . All I'm saying is the base is equal
04:44	to this really special number 2.718 to 8 . And
04:47	then infinite can never write them down because it goes
04:49	on forever to the power of X . And I'm
04:52	gonna draw a graph to show you why this is
04:53	so special in just a second . Now , as
04:56	you know , exponential function , we can take the
04:59	inverse of this function . And from that we can
05:03	get the the inverse of the exponential function . And
05:07	you all know now that the inverse of an exponential
05:09	ysl algorithm , that's what algorithm is . So then
05:13	if you say that the law algorithm with a base
05:16	E because it's the inverse of this guy is given
05:19	by this . We have this come up so often
05:21	a math that we actually give it a new name
05:23	. We don't call it longer than base E f
05:25	X . We call it natural logarithms of X .
05:28	So if you see Ln of X on your calculator
05:31	, what it means is that's just a log a
05:33	rhythm with this very special number E as a base
05:36	. Don't get so worried about the fact that he
05:37	is a decimal . You can have any number you
05:40	want as long as it's positive to be and you
05:42	want to stay away from the number one . We
05:43	talked about why one can't be a base either ,
05:45	but any positive number other than one can be a
05:47	base decimals can be a base fractions can be a
05:50	base . And this is just so happens to be
05:52	international number and it has very special properties that cause
05:56	it to pop up in nature so much . So
05:59	why is he important ? Because we define this exponential
06:02	function . This pops up everywhere in nature and because
06:04	this pops up then it's inverse . The natural log
06:07	or log base E pops up . We never ever
06:09	say log base E . We always just call it
06:12	natural log rhythm of X . Now again , I
06:14	want to do a little bit of a recap .
06:15	Where does the number this number come from ? Well
06:18	if you look at the uh exponential growth of money
06:23	, then we look at the compound interest formula and
06:26	we start compounding more and more and more often and
06:29	so on . And what comes from that is the
06:31	definition of E if you look at a text book
06:34	, it will be defined like this . It'll be
06:36	what's called the limits as in approaches infinity . Don't
06:40	let this word limit scare you . It's actually really
06:41	simple to understand of the following thing . One plus
06:45	one over N . To the power of end .
06:48	Now when you first look at this equation it looks
06:50	like what is this ? This doesn't this doesn't make
06:53	any sense at all . Now what I did is
06:56	I broke that out into a previous lesson in the
06:59	previous lesson . I go in excruciating detail exactly where
07:02	that comes from . That comes from the compound interest
07:05	formula . We've been doing problems with compound interest .
07:08	If you remember one of the , one of the
07:10	parts of the compound interest formula was . How often
07:13	are you compounding it ? That was what we called
07:15	in . Right . Are you compounding yearly than in
07:18	is one ? If you're compounding quarterly in is 44
07:21	times a year . If you're compounding every day ,
07:23	then in is 365 because you're doing 365 compounding periods
07:28	. But in the course of mathematics development people started
07:31	looking what happens if you start compounding every nanosecond ?
07:34	What about every incredibly tiny fraction of a second ?
07:38	What about if you compound all the time ? What
07:40	if you slice the year up into so many little
07:42	pieces that I'm compounding that interest rate all the time
07:46	. In that case the interest , the number of
07:47	compounding periods would go to infinity . You can never
07:50	actually reach infinity , but you can approach infinity .
07:53	That's what this means . And then this is really
07:55	the compound interest formula . So if you start plugging
07:58	in numbers here , for instance , if you put
08:01	the N . Is equal to 1000 into this equation
08:06	, grab a calculator one over 1000 plus one and
08:09	raise that to the power of 1000 . Then what
08:11	you're gonna get is 2.71692 All right . If you
08:16	make and get closer to infinity 10,000 , then the
08:20	number that you're gonna get is 2.71815 . If you
08:24	say it is going to be equal to 100,000 ,
08:28	Then you get 2.71827 . You see it's approaching a
08:32	number as N gets bigger and bigger and bigger ,
08:35	this number doesn't change very much at all . And
08:37	so what ends up happening is when in the what
08:39	we call in math in the limit , that means
08:42	you can never get to infinity . But if I
08:44	sat in front of my calculator and kept putting bigger
08:47	and bigger and bigger numbers for n until the end
08:50	of time I would have enormous numbers . But this
08:53	number wouldn't be changing very much and eventually it would
08:55	reach a limiting number when that limiting number is what
08:59	we call E and it's approximately equal to 2.71828 dot
09:04	dot dot . The reason the dot dots are there
09:06	is because the decimals go on and on forever .
09:08	Non repeating . So getting closer and closer and closer
09:11	. But this is the actual value of you .
09:14	This is what you will see in a algebra book
09:17	and a calculus book in a trig book . And
09:19	it all comes from the exponential growth of money and
09:22	looking at the compounding periods . We discussed that in
09:24	great detail in the last lesson now because this number
09:29	is so special , then we give it us own
09:32	letter and we define an exponential number with that base
09:35	. And because of that we can define the inverse
09:37	of that , which is a law algorithm with that
09:39	base as well . Let's take a second to sketch
09:43	out uh sketch a couple of graphs . What I
09:46	want to do is sketch this exponential formula uh graph
09:50	talk about the characteristics of what's so special about it
09:52	. And then on another graph , I want to
09:54	sketch this natural algorithms so you can see what the
09:56	shape of that looks like and what's so special about
09:59	that . At the conclusion of all of that ,
10:01	then we'll have enough under our belt to solve a
10:03	couple of quick problems with using with the natural algorithm
10:06	. But the punch line is you just use the
10:08	regular laws of logarithms . It's just another algorithm with
10:11	a base E . It's just one that you're never
10:13	gonna be able to shake going here and here .
10:15	Here On out from math . We're never gonna get
10:17	away from natural algorithms or base E exponential . They're
10:20	never ever gonna go away . You will constantly be
10:22	using them . All right . So what I want
10:25	to do is do one graph up here in one
10:27	graph below . And I'm gonna do it freehand ,
10:29	which could be a disaster . But I'm gonna try
10:31	to do it in a way that you might graph
10:33	it yourself because , you know , computers are great
10:37	. I love using computers , but sometimes it's just
10:39	nice to grab uh to grab a piece of paper
10:42	and sketch it out . So we know that this
10:44	thing is F of X . What I want to
10:46	graph is E to the power of X and E
10:48	s 2.718 to 8 . And then a bunch of
10:51	decimal points after that . So if I put the
10:54	number zero for X in there then I'm gonna get
10:56	you to the zero which is one . So I'm
10:58	gonna call this the number one and I know that
11:00	I have a dot right there . It always crosses
11:03	at the number one . And if this is my
11:04	scale then here's the number two and then here's the
11:07	number three and I can even go up to here
11:08	being the number four . Okay ? So I know
11:11	I have a dot through here , I know this
11:13	is an exponential . I know it's gonna go through
11:15	this point and it's gonna go up but we need
11:17	probably one more point in order to get it right
11:19	. So what I'm gonna do is look at the
11:21	number will put one here will put two here .
11:24	Three here , we'll put four here . What if
11:27	I put the number one in here then ? E
11:29	to the power of one E . To the power
11:31	of one is just E . But I know the
11:33	about 2.7 2/7 roughly about here . So I'm going
11:37	to say that this point is basically somewhere right around
11:41	there . Is this exact No it's not . But
11:42	it is trying to be the number E right here
11:47	, Which is 2.71 , 8 - eight or whatever
11:50	. So here's my two points . It needs to
11:51	go through here and it needs to go through here
11:53	. So let me go ahead and switch colors an
11:55	attempt to draw this , it's probably not gonna look
11:58	too good . I'm sorry about that . I'm just
12:00	gonna lips already messed up . I'm gonna try to
12:02	go through there as best I can hugs . The
12:05	the guy right down here goes through hooks north and
12:09	then something like actually I'm pretty happy with that .
12:11	That's pretty close to what the actual exponential curve looks
12:14	like . Maybe there's a little , it's a little
12:15	sharp right here , but you get , the idea
12:17	goes through here and it goes through here . Now
12:19	. The question is , or the thing that we
12:22	want to talk about is why is this so special
00:0-1	?

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