Logarithms - What is e? | Euler's Number Explained | Don't Memorise - Free Educational videos for Students in K-12

Logarithms - What is e? | Euler's Number Explained | Don't Memorise - By Don't Memorise

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00:02	What is E . The two most common logarithms used
00:06	are locked to the base 10 and locked to the
00:08	base e . Why are they the most commonly used
00:11	logarithms ? How do we really understand them ? Look
00:15	to the base 10 is kind of intuitive . It's
00:19	easier to talk in multiples of 10 10 . 100
00:23	1000 and so on . Look to the base 10
00:27	gives us an easier skill to work with Log ,
00:30	tend to the base 10 is one Log 100 to
00:33	the base 10 is too 1000 to the base 10
00:37	is three . And we can see that the multiples
00:40	of 10 can be managed with the skill of natural
00:42	numbers . But look to the base E . Is
00:46	what I'm deeply interested in . It's called a natural
00:49	log and it's also written as Ellen . Yes ,
00:53	log to the base E . Is also written as
00:55	Ln So log 20 to the base E . Can
00:59	be written as Ellen of 20 . The natural log
01:02	of 20 . What does this ? E . Some
01:05	call it a magical number . Some call it an
01:08	irrational constant . Some call it a jeweler's number .
01:12	But the harsh truth is this very few people actually
01:16	understand what he is to get to eat . We
01:19	first need to understand growth , let's say a particular
01:22	thing doubles every time period on this timeline . This
01:26	is today and every unit is one time period .
01:31	Assume you have a dollar with you At the end
01:33	of the first time period . This dollar doubles and
01:37	you have $2 At the end of the second period
01:40	, $2 double to become $4 . And at the
01:44	end of the third period we have $8 . How
01:47	do we look at this growth first We see that
01:51	the numbers at the end of the time periods are
01:53	powers of two . It's in the form to to
01:56	the power X . Where X is a non negative
01:59	integer . To raise to one . To raise to
02:02	to to raise 2 , 3 and so on .
02:05	This was one way in which we looked at double
02:08	growth . Another way to look at it is that
02:11	there is 100 growth every time period one plus 100
02:16	of one gives us to $1 was increased by 100
02:21	to get to . In the second time period $2
02:25	grow to $4 . two plus 100 of two gives
02:29	us four and so on . So doubling the value
02:33	is the same as a 100 increase . So to
02:37	raise two X can also be written as one plus
02:40	100% race to X . It's like you're getting a
02:43	100 return on your investment . But hold on .
02:48	We are making an assumption here , we are assuming
02:51	that growth happens in a discontinuous fashion . We are
02:55	seeing growth in steps here . What about the time
02:58	in between two time periods ? We are seeing no
03:01	growth in between , no growth . And it suddenly
03:04	doubles again . No growth and suddenly doubles . But
03:08	hey , that's not how nature functions everything . Or
03:12	every kind of growth happens gradually . If your high
03:16	today is four ft , you suddenly won't be five
03:19	ft a year later , your height gradually grows .
03:23	When we started , we used to get around 30
03:26	views a day . And after a year we started
03:29	getting around 4000 years a day . It doesn't mean
03:33	a view count just jumped one fine day , it
03:35	gradually increased . So growth in nature is never really
03:39	discreet or discontinuous . Let's see how it really works
03:43	. We take the example of a dollar growing over
03:46	one year at a 100 growth rate . First ,
03:50	we look at the annual growth based on what we
03:53	saw at time zero , we would have $1 And
03:57	at the end of the year we will have $2
04:00	. This doesn't seem right because all the interest cannot
04:04	appear on the last day to make it slightly better
04:07	. Let's divide the year into two equal parts six
04:10	months and six months , Splitting that 100% . The
04:15	growth would be 50 in the first year and 50
04:18	in the second . It would look like this .
04:21	Our initial dollar earned 50 interest in the first half
04:25	to give 50 cents more . Now , what happens
04:28	in the second half ? $1 50 remains as is
04:32	The growth is 50% . So 50 of $1 will
04:37	be 50 cents And this time the 50 cents also
04:41	earned a 50 interest . That will be 25 cents
04:45	. This 1.5 is the sum of our original dollar
04:49	and the 50 cents we meet here . So at
04:51	the end of the first year we have our original
04:54	dollar , then we have the dollar that made our
04:56	original dollar made . And we also have the 25
05:00	cents that these 50 cents made . A total of
05:03	$2 25 . This is better than doubling . If
05:08	we want to understand this , using a formula ,
05:10	it would be one plus 100 over to the whole
05:14	squared . We had half the growth rate over two
05:17	time periods . This is also referred to as semi
05:21	annual growth . Let's push ourselves further . What if
05:25	we had four equal time periods in a year ?
05:28	We have divided one year into four quarters . This
05:31	is how it would look , looks messy , but
05:34	it's actually very simple . If you've understood the concept
05:37	, It's 25 growth every quarter , the formula would
05:42	change to one plus 100% over four . The whole
05:45	rest of four . We would approximately get $2.441 at
05:51	the end of the first year . I suggest you
05:54	pause the video and understand the quarterly growth diagram .
05:57	Really well , The 100 is nothing but one if
06:03	two time periods , then we have to hear If
06:06	four time periods then for here . So the formula
06:09	for end time periods would be one plus one over
06:12	and the whole rest to end clearly more . The
06:16	number of time periods higher will be the returns .
06:20	This will give us the dollar value at the end
06:23	. I probably know what you're greedy . Brain is
06:25	thinking , is it possible to get unlimited money ?
06:29	Let's make a table now . Number of time periods
06:33	and the dollar value in the end . If it's
06:35	just one time period , the dollar value is too
06:39	If two time periods then $2.25 . If four time
06:43	periods then $2.44 . If I divide the year into
06:48	12 equal time periods , my return will be higher
06:52	than this . If I divide it into 365 equal
06:56	time periods it will be even higher . This tells
06:59	us that a dollar at the start of the year
07:01	will become these many dollars at the end of one
07:04	year . If the number of time periods is 365
07:08	, so if we increase this number significantly , that
07:12	is if we increase the number of time periods significantly
07:15	, will this number also increased significantly ? Okay ,
07:19	here are a few more calculations . The number of
07:22	time periods here is one million . Notice that the
07:25	returns improve ? Yes , but they converge around a
07:28	value which approximately equals 2.718 . And that is your
07:34	beloved E . We can't get infinite money after all
07:38	. What would be a layman friendly explanation for Eden
07:42	? It is the maximum possible result after continuously compounding
07:46	a 100 growth over one time period . Yes that's
07:51	e Don't forget we had assumed a 100 growth here
07:56	and that's what he is . It's the maximum we
07:59	get after 100 continuous compounding growth over one time period
08:04	. Notice what compounding does ? The first result is
08:08	100 without compounding , $1 would become $2 . But
08:14	after continues compounding , $1 will become $2.718 . Approximately
08:20	. That would be a growth rate of 171.8% .
08:25	That's like the maximum growth we can have . So
08:28	e . is approximately 2.718 . It's an irrational number
08:33	, which means the digits after the decimal point .
08:36	Do not repeat and go on forever . Just one
08:40	last question . What if the growth rate and the
08:43	time period change ? Will he still help us ?
08:46	Absolutely . There's no problem at all . In general
08:50	, the growth after continues compounding is given to us
08:53	as each of the power are times T where r
08:56	is the rate and T . Is the number of
08:58	time periods . So if we have a 200 growth
09:02	for five years , then it would be defined as
09:05	each of the power . Two times five . We
09:08	squared e to include 200 growth and we raised it
09:12	to five as there are five time periods . It
09:15	will give us each of the power 10 . He
09:18	is nothing but the maximum possible result after continuously compounding
09:23	a 100 growth over one time period .

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What is e? What is Euler's Number or Euler's Identity? What is the Natural Logarithm or logs? what is a logarithmic function? Watch this logarithms tutorial to know the answers to all these questions also learn how the value of Euler's Number is calculated.

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