06 - Proving the Logarithm (Log) Rules - Understand Logarithm Rules & Laws of Logs - Free Educational videos for Students in K-12

06 - Proving the Logarithm (Log) Rules - Understand Logarithm Rules & Laws of Logs - By Math and Science

Transcript


00:00	Hello . Welcome back . The title of this lesson
00:02	is proving the laws of logarithms . So up until
00:06	this point we've actually been using these laws of logarithms
00:09	. There's three primary ultra important rules of logarithms .
00:13	Laws of logarithms . We've been using in the solve
00:15	algorithmic equations and simplify expressions , but I never proved
00:19	them to you and I want to take a minute
00:20	to prove them to you because I don't like telling
00:23	you , hey , this is true . Trust me
00:25	in general , I don't like to , you probably
00:27	could skip this section if you want to but I
00:29	think it's probably a good idea that you watch it
00:31	just so you can know where they come from .
00:32	And also because it's a good exercise in working and
00:36	manipulating the algorithm . So you get your skills uh
00:39	kind of like you get your skills sharpened as well
00:41	. So at the end of this I really like
00:42	you to prove them all yourself . Even after you've
00:45	seen me do it , there's a lot of use
00:46	in that . So we're gonna prove three laws of
00:49	logarithms here . The first one we're gonna prove is
00:51	the following one approve the one that we use all
00:54	the time when we use all of these . But
00:56	this one probably use the most log rhythm base B
00:59	of the product M . Times in . And we
01:04	now know because we've been using this so much that
01:05	that's the same thing as log base B of M
01:08	plus log base B of N . So why is
01:12	it that when you have two things multiplied in the
01:15	log rhythm operating on it ? Why does that become
01:18	the addition of two logarithms ? It's almost like multiplication
01:22	. When you're dealing with logarithms , you take the
01:23	log of multiplication . It kind of gets transformed into
01:27	addition of law algorithm . So the multiplication operation becomes
01:31	addition . Which is very useful for more advanced math
01:34	when we're trying to simplify things . Sometimes , multiplying
01:37	things gets tricky and dicey . You can change it's
01:39	the addition . If you just take the law algorithm
01:41	of the thing . So we're trying to prove this
01:42	thing here . Also one more thing I'm going to
01:44	say probably in the beginning you might say , oh
01:47	, that proof was neat , but I never would
01:48	have thought of that . That's okay . None of
01:50	us think of the stuff to begin with , right
01:52	? You have to see it right ? As long
01:54	as you understand what I'm doing , that's all I
01:55	care about . I don't expect you to know how
01:57	to prove this . I don't expect you to feel
02:00	like you should already know how to do this .
02:02	I just want you to follow it . That's all
02:03	I want you to do . So that you can
02:04	sharpen your skills . All right . So in order
02:08	to do this , we know we want to add
02:10	these guys together . So to make things easier ,
02:12	let's let the following things be true . Let's let
02:16	some new variable X . Equal the log base B
02:20	of M . And let's let a new variable .
02:22	Why be log base B of N . You might
02:26	say why am I introducing new variables ? Well ,
02:28	that happens all the time . In proofs . The
02:30	proof is like it's like a blank canvas for you
02:33	to paint on . You can paint anything . I
02:34	can paint what I want , I can paint jupiter
02:37	or Saturn , I can paint cows or chickens .
02:39	It's whatever I want to do in the proof ,
02:42	I can do what I want as long as it's
02:43	mathematically legal . So I'm just gonna let the variable
02:46	X equal to this and this equal to this .
02:48	That's fine . The rest of it . I'm gonna
02:50	try to logically connect the dots and make this thing
02:52	be true . Okay . But if I let these
02:56	things true , then the following things are true .
03:00	Because this is a law algorithm , then I then
03:02	no , because of this first line , B to
03:05	the power of X is equal to M . B
03:08	to the power of X . M . That's the
03:09	definition of a log rhythm . And also B to
03:13	the power of Y is equal to n . So
03:16	because of what I let these variables equal then these
03:19	variables are equal . You might say this is not
03:22	at all looking like that . That's fine . That's
03:24	what I said . You probably you know , unless
03:26	you're a math guru , you probably wouldn't know to
03:29	do this . I mean none of us would but
03:30	as long as you can follow what I'm doing ,
03:32	that's all I care about . All right then .
03:36	Or I should say thus . Use words like thus
03:39	. And a proof . If I let this true
03:41	, then this becomes is true because of the definition
03:43	of the law algorithm , then I know I want
03:45	to multiply these things together . M times in .
03:48	Okay . But now I know what M and N
03:50	. Are they came from these definitions ? You see
03:52	? Now I have equations for what M and N
03:54	are . So then M times in is this times
03:58	this which means that's B to the X , times
04:00	B to the Y . You see ? Because I
04:02	know what imminent are those things are multiplied together .
04:05	But because these have the same base in different exponents
04:08	, I can then say that this is B to
04:10	the power of X plus . Why ? Because I
04:12	can add these exponents anytime I have the same base
04:15	, I can take uh those exponents there . I
04:18	can I can add those exponents . Now . Ultimately
04:21	I have M times in . And I'm trying to
04:23	take the law algorithm of that and I want to
04:24	see what it equals . Now , I have an
04:26	expression for what M times N actually is . So
04:30	in order to figure out if this is gonna work
04:33	out , I'm gonna take long algorithm of both sides
04:35	. I'll take log of base be both sides because
04:42	I can do anything . I want to an equation
04:44	as long as I do it to both sides ,
04:46	I'm gonna work with this one . I'm gonna say
04:47	log base B of M times N is equal to
04:52	. I'm gonna have to take the log of this
04:53	log base B . Of B to the power of
04:56	X plus Y . Now this B to the power
04:57	of X plus Y . This is what the law
04:59	algorithm is operating on . But remember we talked about
05:02	90 million times about the idea that a law algorithm
05:06	with the base B is the inverse function of an
05:10	exponential with the base be there in verses . So
05:12	when they operate on each other , they annihilate each
05:14	other . That is what the definition of the inverse
05:16	. Does the inverse function under does what the parent
05:20	function or what the other function did in the first
05:22	place . So since we have an exponential here ,
05:24	it raised to the power B to the power of
05:26	this . But the law algorithm um does that .
05:28	So on the right hand side the log rhythm cancels
05:31	with the base B . And all I have left
05:32	is X plus Y . On the left hand side
05:36	I have the law rhythm based B . Of M
05:38	times N . You might say great but I don't
05:41	care about X plus Y . But then you remember
05:43	wait a minute . I define what X and Y
05:45	were up here . This X and Y are the
05:48	same ones that are rippling through the problem statement .
05:50	So then you know that the law algorithm based B
05:53	of M times N is equal to X . But
05:56	X is the log rhythm based be of them log
06:00	base B of M . There's a plus sign here
06:03	. Why why is log base B events ? Yes
06:07	. And if you look back and compare these things
06:10	, these are exactly the same thing . Alright ,
06:14	Because log base B of the product becomes the algorithm
06:17	of the first one . Plus the algorithm of the
06:18	second one . That's exactly what we set out to
06:20	prove . Do I expect that you would have known
06:22	how to do that ? No . Most teachers are
06:24	not going to tell you prove the first law of
06:26	algorithms , but what does it do ? It allows
06:29	you to understand how to get this from the definition
06:32	. That's a property of logarithms . When you multiply
06:35	them , you add exponents when I take the log
06:36	of both sides , the annihilation that happens here .
06:38	That's a property of algorithms . So in the process
06:41	of learning how to prove these things , you actually
06:43	sharpen your skills with dealing with logs . That's really
06:45	all I care about in this lesson . Okay ,
06:50	lets go and prove the the one about exponents .
06:54	Next . We're gonna prove log base B of M
07:01	. To the power of K . Some number to
07:03	the power of K . And we know that that's
07:06	equal from our laws of logarithms to K . Log
07:10	be of em like that . Okay . K log
07:14	b . Bass beat of the number M . So
07:17	we basically , we can take an expanded , we
07:19	can bring it out in front of the long room
07:20	, that's all this is telling us . How do
07:21	we prove that ? So we need to do ,
07:23	we're gonna do the similar kinds of assumptions . Okay
07:26	, We're gonna have to let some things happen ,
07:29	okay . In this case we're gonna let we're gonna
07:33	let X equal to log B . Of N .
07:39	Okay , now we could let Y equal to log
07:42	B . Of N . But you see in this
07:44	property of logs there is no end anywhere else .
07:46	So we don't really need to define that . Here
07:48	, we had an M . And an end .
07:49	So we had to kind of have a term for
07:51	each one of them . But here there is no
07:53	end anywhere , so there's no reason to really do
07:55	that . But from this single definition here , something
08:00	pops out . The same exact thing is before B
08:03	. To the power of X is equal to him
08:06	. So so far , it's just a rewrite of
08:07	what we did on the other board . But we
08:12	have a conclusion that we can draw here , I
08:16	should say . Thus , right ? Ultimately what I
08:20	want , I want him to the power of K
08:22	. Okay , into the power of K . So
08:24	this thing um to some power of K is what's
08:29	on this side , B to the x rays to
08:31	the some power of K . All I've done is
08:33	I said , hey , this gives me a definition
08:35	for what I am , is I want to raise
08:36	it to a power because that's what's in my proof
08:38	. So I raise it to a power . So
08:40	I have to raise this side to the power because
08:41	that's what it's equal to . But this means that
08:44	M to the power of K is B to the
08:48	how do I want to write it ? K .
08:49	X . Because this is an exponent raised to an
08:52	exponent . So I'm going to multiply those things together
08:55	. All right . And what do we do in
08:57	the last uh thing once we had it all kind
09:00	of written out like this we wanted to get em
09:02	times into to match our problem statement here , we
09:05	have M to the power of K . To match
09:07	what's in our proof . So now we want to
09:08	take log of both sides . So we'll take the
09:10	law algorithm . Uh We'll take a base be law
09:13	algorithm of M . To the power of K .
09:15	La algorithm based B . Of what's on the right
09:18	hand side , B . To the power of K
09:20	. X . Like this . But then , you
09:22	know that on the right hand side you're gonna get
09:25	the same annihilation here , you have a log of
09:27	an exponent . Those two things are gonna cancel just
09:30	like we had a log of an exponential here ,
09:33	we have a log of an exponential . here on
09:34	the right hand side . All you're gonna have is
09:36	K times X . On the left hand side .
09:38	You're gonna have log base B of M . To
09:41	the power of K . You have K times X
09:44	. And you say , well wait a minute ,
09:44	I know what X . Is X . Was log
09:47	base B of M . So you're gonna have log
09:50	base B of M to the power of K .
09:53	Is K , multiplied by X . But I know
09:55	what that is . It's log base being of them
09:59	. You see you take the law algorithm of something
10:01	with an experiment , you can just take that exponent
10:03	right out front and then multiplied by the log of
10:05	of that of that base there . So that is
10:08	the proof of the one of the three laws of
10:11	logarithms . The one that deals with exponents . Now
10:13	the one that I say for last is the one
10:15	that looks kind of like this . Remember there's one
10:17	that talks about multiplying things becomes addition of logs .
10:20	And then there's another one that talks about division of
10:23	things become subtraction of logs . So our last one
10:26	that we're gonna do this will be the last thing
10:28	will prove in this lesson , isn't want to prove
10:31	. Let me flip my page here just to make
10:33	sure I don't uh mess anything up . Let's prove
10:38	this last one , proof the following thing . Log
10:43	base B . Of the division M over N .
10:47	Is equal to log base B . Of M minus
10:51	sign . Log face beat in . So I want
10:54	to do the subtraction there . So ultimately this proof
10:58	is going to be very very similar to what we
11:01	did before . But let me show you how we're
11:05	going to get there . The first thing we're gonna
11:07	do is we're gonna recognize that this statement right here
11:11	. It's really the same as log base B .
11:14	Of Mm multiply by one over n . Right ?
11:19	Multiplied by one over N . That's how you get
11:21	a division here . You multiply by one over N
11:23	. And since we know what happens when we multiply
11:26	things , we can use a similar proof to go
11:30	. In fact , we can even use that thing
11:31	. We could just substitute in and use it .
11:33	But that's no fun . Let's do it . Let's
11:34	do it the real way . Let's let's do it
11:36	the right way . Let's let the following things happen
11:41	. Let's say let's let X . Equal to the
11:43	log base B . Of something called him . And
11:46	we'll let y equal log Base B . of something
11:50	called one over in . Mhm . One over end
11:54	like this . Why am I doing that ? Because
11:56	I know I'm gonna need an M . And I
11:57	know I'm gonna need a one over N . It's
11:59	different than the other one day and they were just
12:00	multiplied together . I'm going to need some fraction with
12:02	an M . On the bottom here . Mhm .
12:05	Okay . So we'll do the same thing again .
12:08	Then the following things are true . B . To
12:12	the power of X . Is in And then from
12:16	this one B . to the power of why is
12:19	one over in . So we're gonna need those .
12:24	Alright now , ultimately what I'm trying to do is
12:25	I'm trying to form the product of these things so
12:27	I can take the law algorithm , right ? So
12:29	what I'm gonna have is M . Times one over
12:33	N . Because I'm trying to make him over in
12:36	is M over in . But I know what these
12:38	things are equal to its B . To the eggs
12:41	times B . To the Y . Because that's him
12:44	. And that's one over N . Okay . But
12:46	what is this B . To the X . Plus
12:49	Y . So what I figured out is M .
12:51	Over in is exactly the same thing as B .
12:55	To the X plus Y . So I'm gonna take
12:59	and write that down separately . Em over in is
13:02	B . To the X . Plus . Why ?
13:04	And now I want to take the algorithm of both
13:05	sides . Why ? Because I want the law algorithm
13:07	of this quotient of this division . That's what I'm
13:09	trying to get to . I'm trying to take the
13:11	algorithm of that . So I had to kind of
13:12	do all of this to get to where I had
13:14	an expression for this . And so I'll take the
13:16	log rhythm of both sides . So it has to
13:18	be a base be log rhythm base B of em
13:21	over in equals log base B . Of B .
13:27	To the X . Plus . Why ? all of
13:28	this is wrapped up inside that longer than I have
13:30	the same thing happening on the right The law algorithm
13:32	of an exponential annihilate each other . All I'm going
13:35	to have on the right hand side is X plus
13:39	Y . On the left hand side . I'll have
13:41	log base B in over in like this . Mhm
13:48	. All right . So what am I gonna have
13:49	next ? I know what X and Y are .
13:52	So let's substitute for that log base B mm over
13:57	in is equal to X plus Y . But I
14:00	know it . X and Y . Are X .
14:02	Is log base B log of him log base B
14:07	. Of M . And then the why is this
14:09	guy log base B one over in . So as
14:12	plus sign from this log base B one over in
14:19	. It's hard to read . I'm sorry about that
14:20	. This is a capital in right here like this
14:23	, so X plus Y . So I'm just plugging
14:25	what I have now you think well this isn't quite
14:27	right , this is a plus sign , this is
14:29	a minus sign , it's not quite right , but
14:30	we're gonna get there , how are we gonna get
14:32	there ? Let's go . Hm . Well if you
14:35	think about it , let me go over here .
14:36	Let me just write a note over here . This
14:39	is the same thing as the law algorithm Base B
14:42	of N to the -1 . How do I know
14:45	that ? Because one over and is the same thing
14:47	as into the -1 . So this is an exponent
14:50	. So this exponent can come out in front here
14:52	. So then I have logged base B of em
14:56	over N equals log bass beat of M . And
15:01	because of this the exponent could come down , it's
15:04	gonna be a minus sign log and then you have
15:08	an end right here and this is exactly what we're
15:11	trying to prove . The log of . The division
15:13	of N . M divided by in is the log
15:15	of M minus the log in . It's exactly what
15:18	we have right here . It all comes from this
15:19	little experiment and it comes from having to define things
15:22	this way , so I can make the product of
15:24	Eminem , So are the division of M divided by
15:27	N . Do I expect you to know this ?
15:28	No , of course not . The first time you
15:29	see this , nobody knows how to do this stuff
15:31	. All I care about is that you can follow
15:33	through so that it serves two purposes . It sharpens
15:36	your skills so that you know where things come from
15:41	. I mean , you know how to use these
15:42	laws of logarithms to in order to make a logical
15:45	conclusions . And it also shows you that these laws
15:48	of logarithms don't just come from the sky . They
15:50	come from logical progression of thought . We start from
15:54	here , we let this equal to this . Notice
15:56	how nobody , I don't wanna say , nobody on
15:58	this earth . There's some pretty brilliant math guys out
16:00	there , but not very many people can look at
16:02	this and say , oh yeah , I know how
16:04	to prove that , no problem , I can see
16:05	all the steps , almost nobody can do that .
16:07	But what we can say is if we let this
16:09	equal to this and this equal to this , then
16:11	this must be true . Then if we if we
16:13	multiply this in this , it must be this times
16:15	this and then we just ripple through . Every little
16:17	step is an incremental uh step towards the finish line
16:21	, that's all I care that you understand . So
16:23	make sure you understand this . If you want to
16:25	grab a sheet of paper and improve these yourself ,
16:27	it definitely can't hurt you . And then follow me
16:29	on to the next lesson . We're going to start
16:30	to learn more about the applications of logarithms and exponential
16:34	in math .

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