15 - What is a Logarithm (Log x) Function? (Calculate Logs, Applications, Log Bases) - Free Educational videos for Students in K-12

15 - What is a Logarithm (Log x) Function? (Calculate Logs, Applications, Log Bases) - By Math and Science

Transcript


00:00	Hello . Welcome back . The title of this lesson
00:02	is called . What is a law algorithm ? Or
00:05	I could re title this thing , logarithms explained or
00:08	understanding algorithms . This is part one of several lessons
00:11	only algorithms . A law algorithm is one of the
00:14	most important functions in all of science in all of
00:17	math . And I know I say this a lot
00:18	but really I'm trying to emphasize when I'm really trying
00:21	to tell you things that are extremely important . And
00:23	you'll run into logarithms in your chemistry classes . You'll
00:26	run into logarithms and all of your physics classes .
00:28	Any engineering class , any math class , you're going
00:31	to run into algorithms . So it's not every day
00:33	I can teach you about a function that's important .
00:35	I remember the very first time I got introduced to
00:38	using algorithm in a chemistry class , I went up
00:41	to the teacher because the ph scale you've probably heard
00:44	of ph and chemistry has long algorithm in its definition
00:47	. It's the negative log a rhythm of hydrogen ions
00:50	in a solution . That's a crazy , complicated sounding
00:52	thing . But basically it's the law algorithm of a
00:54	number . Right ? And so I went to the
00:56	teacher and I said , what does this mean ?
00:57	Law algorithm of hydrogen ? What does log log mean
01:00	? Right . And she said , well that's a
01:01	button on your calculator . And I love this teacher
01:04	. She's a fantastic teacher . But she didn't understand
01:07	at a fundamental detailed level what algorithm was for her
01:10	? It was just a button on the calculator to
01:12	calculate things . I want to go beyond that with
01:14	you . I want you to understand what algorithm is
01:17	. I want you to know intuitively why we have
01:19	algorithms and why they naturally fall out of math because
01:23	the knowledge that you gain in this lesson will be
01:25	applied to many , many , many courses down the
01:27	road . All right . So before I get into
01:30	what algorithm is I want to give you a little
01:31	motivation . Just two examples I already mentioned the first
01:34	one . The famous ph scale , acids and bases
01:37	is so fundamental to chemistry . It's a log arrhythmic
01:40	scale . So long algorithms come into play anytime you
01:42	calculate anything to do with ph concentrations in chemistry .
01:46	Right . Another very famous example Is the Richter scale
01:50	of earthquake energy . You know , you have uh
01:53	we say earthquake was a 6.2 on the Richter scale
01:56	, earthquake was a 7.3 on the Richter scale ,
01:58	earthquake was 1.2 on the Richter scale . What does
02:01	that mean ? Well , a Richter scale is a
02:03	log arrhythmic scale . So by the end of this
02:05	lesson , I want you to understand what a log
02:07	is , but also why do we use them for
02:08	these scales ? Why don't we just use numbers ?
02:10	Why do we have to use logs ? So by
02:12	the end of this , you're gonna understand all of
02:13	that . There are many other examples of logarithms .
02:17	I could go on and on an electrical engineering .
02:19	We use them to plot frequency response of amplifiers and
02:23	mechanical systems . There's tons of of uses for logs
02:26	. But for now we want to go and learn
02:28	what a log actually is . So , in a
02:30	nutshell , this is what a log rhythm is .
02:33	A law algorithm is the inverse function of the exponential
02:37	function of an exponential function that we've already learned about
02:40	already told you exponentially are so incredibly important . And
02:43	they are . So , the inverse of those functions
02:45	, which we've already learned in verses in the last
02:47	lessons . The inverse of exponential functions is what we
02:50	call a long algorithm . That's why logs are so
02:53	important because exponential are also so important . And they
02:56	go together like peanut butter and jelly basically . All
02:59	right , So that's so important . I actually want
03:00	to write that down . If you pull anything out
03:02	of this , I want you to pull this a
03:04	log rhythm . Yeah . Is the in verse the
03:13	universe function of an exponential function . I don't always
03:25	write definitions out , but this one is extremely important
03:27	. So I'm gonna keep that in the top .
03:29	It's the inverse function . And if you remember what
03:31	is an inverse function . We already talked about that
03:33	in the last lesson , inverse functions go together like
03:36	peanut butter and jelly because inverse functions can undo each
03:39	other . If I have function number one and function
03:42	number two . And we know that they're in verses
03:43	of each other . If I run a number through
03:46	the function , calculate the answer , take that answer
03:49	and put it through the next function and calculate that
03:51	answer . The number I get out is going to
03:53	be exactly the same as what I started with because
03:56	the inverse function kind of undid or did the opposite
03:58	calculation of what the first function did . So if
04:01	I stick a one onto the input function , get
04:04	the intermediate , run it through the second function ,
04:06	I'm going to get a one on the output .
04:08	If I put a two on the input I'll get
04:10	a two on the output of the inverse . If
04:12	I put a negative five on the input and run
04:14	it through both functions , I'll get a negative five
04:16	on the output . Whatever I put it on the
04:17	on the input , I'm going to get it on
04:18	the output because the inverse undies it . And that's
04:22	not a technical word , it's my word but it's
04:24	a good word to use . So we know that
04:26	a log . So you can use the full word
04:29	log rhythm or you could just use the word log
04:32	, can quote undo an exponential . So remember when
04:43	an exponential function as like two to the power of
04:45	X . Or three to the power of X .
04:47	Right ? If we run the number through that exponential
04:50	function take the output and then stick it through the
04:52	same similar longer than the inverse function , we'll get
04:55	exactly the same thing as we started with because it
04:57	does or does the opposite of that exponential function .
05:01	Right ? One of the biggest biggest biggest uses of
05:04	logarithms is we can use to solve exponential what's equations
05:18	. I hate it whenever you look in a book
05:20	and it explains what something is , but you have
05:21	no idea why it's useful or even if it's very
05:23	important , this one's super important because we can use
05:27	logarithms to solve equations that have exponential in them .
05:30	Why ? Because every equation we solve , we always
05:33	have to do the opposite right ? The opposite operation
05:36	to get the variable by itself . If you add
05:38	something to the variable , you might do the opposite
05:40	, you might subtract , that's by the way ,
05:42	an inverse kind of operation . If you multiply in
05:45	the equation , then you might have to divide to
05:47	get X by itself . If you have a square
05:49	, you might do the opposite the square root ,
05:51	right ? We've been doing this forever , right ?
05:54	But now that we have no when an exponential function
05:56	is , we might often have to do the opposite
05:58	of that thing to get the variable by itself .
06:00	The opposite is going to be what we call the
06:02	law algorithm . So that's why logs , one of
06:05	the reasons why logs are so important . Now ,
06:07	here's our roadmap . I'm gonna write down the definition
06:09	of algorithm , we're gonna work a few very simple
06:12	problems so you know how to handle it . And
06:14	then we're gonna draw some grass to show you that
06:16	the inverse of an exponential really is the algorithm and
06:19	what the graph of algorithm looks like . And then
06:21	at the end of the lesson , I'm gonna go
06:23	into a lot more detail about why we use logarithms
06:26	in the Richter scale and ph and some other applications
06:29	because I want you to understand why we care because
06:32	that's really the point of this thing . So here
06:34	we have enough space . Let's go ahead and write
06:36	down the definition of the law algorithm . So here's
06:39	what a log rhythm is . Al algorithm is ,
06:44	the following thing . And I know it's a mathematical
06:46	were definition . I'm gonna make it very clear for
06:48	you if the number B . This is a letter
06:52	, but we're going to call it a number B
06:54	. And the number in are positive numbers with B
07:04	not being able to one because B is going to
07:06	end up becoming the base of the law algorithm .
07:08	And we already know that the base of an exponential
07:10	can't be one . Because we talked about if you
07:12	have one as a base , you don't have an
07:13	exponential function anymore . But anyway , if we have
07:16	these two numbers be an end , they're both positive
07:18	, but they can't be equal to one . They
07:19	can't be equal to one . Then we say that
07:22	the law algorithm using the base B of the number
07:27	N is equal to K . Right ? And this
07:31	is if well let's do it like this if and
07:36	only if fucking spell only if and only if B
07:43	to the power of K is equal to in .
07:46	So here is the definition of the law algorithm .
07:48	gonna leave it on the board because well , reference
07:50	it a lot . I know it doesn't make a
07:51	lot of sense when you see all this jibber jabber
07:54	everywhere , but I'm gonna make it very clear .
07:56	What I want you to understand first of all is
07:58	that here is kind of a relation that involves this
08:01	algorithm . I want you first to understand that the
08:03	law algorithm has a base here , we put the
08:05	letter B there , but in real life they're not
08:07	letters , the basis of logarithms are just like the
08:10	basis of exponentials for an exponential . Remember the base
08:13	was the number on the bottom two to the power
08:14	of X . The base was to 10 to the
08:17	power of X . The base was 10 five to
08:19	the power of X . That's an exponential . The
08:21	base was five . So for logarithms we have bases
08:24	also . So you might have log rhythm with a
08:26	base two logarithms . With the base five law algorithm
08:28	with a base 10 . Why do you have to
08:30	have basis for logarithms ? Well , it's because it's
08:33	an inverse of the exponential function . So of course
08:35	if the exponential has a has a base to determine
08:38	the shape and if I'm just reflecting that graph over
08:41	to find its inverse , that's what an inverse is
08:43	. Then the inverse , which is the log rhythm
08:45	also has to have a base and it's the same
08:47	base as the exponential function that you are talking about
08:50	. So we have an exponential longer than base here
08:52	. Okay , so what we have is a log
08:55	rhythm with a base of some number equal some other
08:58	number . And the relationship of all this means you
09:00	take the base to the power of whatever you have
09:03	on the right hand side , we're calling it K
09:05	is equal to some number . All I want you
09:07	to know about this definition right now is that law
09:10	algorithms can be transformed into exponential because you have the
09:13	same things here , B N and K be in
09:16	K . And these exponential is can be transformed into
09:19	algorithms , B K and n . B K and
09:21	N . So anytime you have an exponential , you
09:24	can always write it as a law algorithm and any
09:26	time you have a longer rhythm , you can always
09:28	write it as an exponential because they're in verses of
09:30	each other . All right . So let's go through
09:32	a couple of very simple problems that you'll understand very
09:36	quickly . And then we'll ratchet up the complexity as
09:38	we go to me and we'll draw some pictures to
09:40	make sure you really understand . Let's say a real
09:43	life example . Let's say we have the law algorithm
09:46	. If I can spell algorithm the law algorithm ,
09:49	Base two of the # eight , how would you
09:52	calculate this ? Well we have a base to and
09:56	we're taking the law algorithm of eight . Here's what
09:58	you do to translate logarithms . You always have to
10:01	translate them into an exponential because they're basically can always
10:05	be written in terms of exponential . So what you
10:07	do is you say the following thing , Okay ,
10:09	you say the base right ? That to the power
10:13	of some number , I don't know what it is
10:15	. X is equal to eight , two to the
10:19	power of whatever the answer is . That I'm trying
10:21	to calculate here . That's what I'm trying to find
10:23	. Trying to find the exponent is equal to whatever
10:25	the longer than that you have here is . Now
10:27	let me ask you , how would you figure this
10:29	out ? What is the exponent here ? That makes
10:32	two to the power of that exponent ate . Well
10:34	this means that X has to be equal to three
10:36	. Sorry X has to be equal to I can
10:39	write it correctly three . How do I know that
10:41	? Because two to the third power is 82 times
10:44	two times two is eight . So what I basically
10:46	said here and just stay with me is that the
10:49	longer them ? Base two of the number eight is
10:53	equal to three . And the reason I know that
10:55	this thing is equal to three is because Because I'm
11:00	gonna triple underlined it's because too which is the base
11:04	to the power of three . On the right hand
11:06	side of the equal sign equals eight . Two to
11:09	the power of three is equal to eight . I'm
11:10	gonna run my fingers a few times two to the
11:12	power of three is 82 to the power of three
11:14	is 82 to the power of three is eight .
11:16	That is going to be how you translate logarithms .
11:19	Every time you write them down you say base to
11:22	the power of what the thing is equal to is
11:24	equal to eight . The thing you're trying to find
11:26	in a log rhythm . The thing that the log
11:28	rhythm gives back to you is the exponent required to
11:32	calculate this number . The mm said that one more
11:34	time because this is the kind of thing that you
11:36	learn after . You do a lot of problems but
11:38	it's often not really told to you . The law
11:40	algorithm as its output gives back to you what exponent
11:44	is needed to make this thing equal to eight when
11:47	I use this base . That is why logarithms and
11:50	exponents are inverse is look how I took this long
11:52	algorithm and I write it as an exponential expression .
11:55	Two to the power of three is eight . This
11:57	is inequality here too . To the power of three
11:59	is eight . I can also start with this and
12:01	go backwards . I can write this as a log
12:03	rhythm for the same reason , the basis to Of
12:06	the # eight . The exponent three is what's coming
12:08	back . So you're often converting back and forth between
12:12	exponential form and algorithmic form . You have to get
12:15	used to that . So let's do a couple more
12:17	of these to make 100% sure that you are getting
12:20	comfortable with it because it's the most important thing .
12:22	Okay let's say I have the law algorithm base to
12:27	remember this base can be any number other than one
12:29	, but I'm just using to hear of the number
12:31	16 . How would I figure out what the base
12:34	of the algorithm ? Base two , logarithms of 16
12:37	. How would I figure out what that is ?
12:38	Well I have to write a little equation to figure
12:39	out what that is . I know that the base
12:42	of this thing , So the power of something that
12:44	this law algorithm is going to equal is going to
12:47	have to equal 16 because logarithms return the exponents ,
12:51	it returns an exponent back to you . That's what
12:53	algorithms do . And I say it a few times
12:56	. I want you to remember that law algorithms give
12:57	you back an exponent . So two to the power
13:00	some exponent . That's gonna be what this log is
13:02	equal to is equal to 16 . Now if you
13:03	run through it it can't be too squared , it
13:06	can't be two cubed X has to be equal to
13:08	four because two to the fourth power is actually 16
13:11	. So another way of writing this is the log
13:15	Base two of 16 is equal to four Because triple
13:20	underlined the reason this is true is because two to
13:24	the power of this exponent that the log rhythm gave
13:27	me back is actually equal to 16 . You see
13:30	we're following the same recipe in both of these examples
13:33	to to the 32 to the third power eight .
13:36	That means longer than base two of eight is three
13:39	. The logger then gave me back the exponent that's
13:42	needed when I raise to two , that exponent to
13:44	give me this and I can write an exponent form
13:46	like this , This one is two to the power
13:49	of four is 16 . 2 to the power for
13:51	16 . That means the log rhythm gave me back
13:53	the exponent needed . So I raised to the base
13:56	to that expanded to give me that number . So
13:58	it's literally a reverse exponent uh inverse , reverse exponential
14:02	function . The exponential function is the base to the
14:05	power of some of the exponent . The law algorithm
14:08	is like okay here's your base . Here is the
14:11	final number that I want you to kind of have
14:13	tell me what exponent is needed to give me that
14:16	number . That's what the law algorithm does . It
14:18	gives you the exponent back . So you have to
14:21	get the use of saying two to the power of
14:22	whatever is on the other side of the equal sign
14:24	equals this , two to the power of whatever's on
14:26	the other side of the equal sign equals this .
14:28	I'm saying it a lot of times because it's literally
14:31	the most important thing in this whole lesson . Right
14:34	, let's take a look at another one . What
14:35	about the log Base two of the # 1 ?
14:39	What would that equal to ? Well , I'm gonna
14:41	show you that that's going to be equal to zero
14:43	. How do I know it's equal to zero ?
14:44	Well it's because of the following thing because the base
14:48	to the power of this exponent that the law gave
14:51	me back is equal to whatever I'm taking the longer
14:54	the love . And I know that this is true
14:56	because anything to the zero power is one right ?
14:59	Make sure you understand that every time based to the
15:02	power of this is equal to this . That's exactly
15:05	what I wrote . What if I have the law
15:07	algorithm based too Of the number 1/2 Is equal to
15:12	-1 . How do I know this is true ?
15:14	Well I know it's true because the base To the
15:18	power of -1 is going to be equal to whatever
15:22	I was taking the longer rhythm was remember the longer
15:24	than gave me the power back . And you know
15:26	that this is true because two to the one half
15:28	means I can drop it in the denominator and make
15:30	it a positive power right ? And so if I
15:33	want to generalize this whole thing two to the power
15:36	of three is 82 to the power of four is
15:38	16 . 2 to the power of zeros . 12
15:41	to the power of negative one is one half .
15:43	I can generalize it and I can say that little
15:44	log base two of the number N . Is equal
15:50	to K . And that's because to to the exponent
15:54	K is equal to whatever this number I was taking
15:57	the log rhythm . What of and don't forget that
16:01	this base in this . In all of these cases
16:02	I'm using the number two but the base can be
16:04	three . The base can be fore , the base
16:06	can be 10 . The base can not be negative
16:09	though because remember we go back to our definition if
16:12	being in er positive numbers but the base can't be
16:14	one . Why can't the base B . One ?
16:16	The base can't be one because if you have one
16:19	as the base of an exponential then you don't have
16:21	an exponential at all . One to the power of
16:23	anything is just one and you can't have negatives for
16:25	the base . For the reasons we talked about in
16:27	the exponential function because you don't have an exponential function
16:29	there either go back to the lesson on exponentially if
16:32	you forget that . So the base has to be
16:34	the same kind of base as an exponential function has
16:36	to be positive and the base can't be one .
16:39	So as to be bigger than zero but it can't
16:40	be one . Now . This definition should make a
16:42	little more sense . The log base B of the
16:45	number in as K . If and only if B
16:47	to the power of K is in B to the
16:49	power K is in exactly as we've done it with
16:51	numbers down here . Okay , so so far I
16:54	have shown you the mechanics of what you do when
16:56	you see a log rhythm on the page . All
16:58	you have to do is say the base which is
17:01	written right underneath the log rhythm to the power of
17:03	whatever the thing is equal to is equal to whatever
17:07	. Uh you're taking the log rhythm of because the
17:10	algorithm gives you the exponent back of whatever you're taking
17:12	the log rhythm of , right . But we haven't
17:15	really done anything graphical . We haven't really shown you
17:17	that their inverse functions . We haven't really done any
17:19	of that . So what we need to do now
17:22	and draw a couple of quick pictures to show you
17:24	that the law algorithm is really the inverse function of
17:27	an exponential function . And then you'll really understand more
17:30	about why they're exactly so closely related cousins of one
17:33	another . Okay , so why is this behavior happened
17:37	? Why is it that two to the power of
17:39	K ? Is this why is it that two to
17:40	the power of zero ? Is this why is that
17:42	the case ? Let's look at the following thing .
17:45	Let's draw two graphs . This is the middle of
17:47	my board . So I'm gonna try to draw two
17:48	graphs . I'm gonna try to draw the exponential function
17:50	right here and then I'm gonna try to draw over
17:54	there . I'm gonna try to draw the algorithm .
17:58	So here I have F of X and here I
18:01	have X . And I want to draw first the
18:03	exponential function . So what I'm gonna draw is my
18:06	exponential function is F . Of X . Is to
18:09	to the power of X again . Remember I'm using
18:11	the base of two but this base can literally be
18:13	10 or five or 15 , it can't be negative
18:16	and it can't be one , but it can be
18:17	anything else . It can even be decimals . 1.3
18:19	to the power of X is perfectly fine as a
18:22	base for this exponential function . But two is very
18:24	easy because we can calculate things in our heads really
18:26	, really easy with the number two . Now ,
18:28	I think I need to um draw some tick marks
18:33	and you try to be kind of precise with them
18:35	. So here's one , here's to here's three ,
18:37	here's 12345678 Okay , I barely made it . That's
18:42	as many as I need . Remember . Every exponential
18:45	uh with a positive base like this a bigger than
18:48	one starts off like this and goes up to the
18:51	right , so we know this exponential function is gonna
18:53	look like this , we know it's going to go
18:54	through the value of one because this is one right
18:56	here , right ? So we know it's going to
18:58	go through here , but we need to be a
18:59	little bit more precise so that when we draw the
19:01	inverse it'll make a little more sense . So what
19:03	we're gonna do is we're gonna calculate um zero comma
19:08	one , that's going to be a value here .
19:10	And then when we put the value of one in
19:13	here , this is one , this is two ,
19:15	this is three , we put the value of one
19:17	in here . Uh to to the one power is
19:19	going to be too , so it's going to be
19:20	up right here at two . Right now we put
19:24	two in here , it's gonna be two to the
19:26	power to is four . So here's 1234 So it's
19:28	gonna be something like this . Okay , And then
19:32	three so two to the power of three is going
19:34	to be eight . So that's why I needed 82345678
19:37	It's gonna be eight and it's gonna be something kind
19:40	of like this . All right . So you see
19:43	it's not a straight line . If it was a
19:44	straight line , it would be like this . But
19:46	you see it's curving so we need to do is
19:48	attempt and I'm probably gonna mess this up . But
19:51	it needs to go something like this needs to go
19:53	through this point , through this point , through this
19:55	point and then up . It's not perfect , but
19:58	I'm not gonna erase it because it's actually close enough
19:59	. It goes through all of these points . Now
20:01	I need to label some points because I'm going to
20:03	draw the same points over in the log arrhythmic curve
20:05	. Okay , what do we have right here ?
20:07	Well , this point goes over here , it's two
20:10	to the power of three . That was equal to
20:12	eight . Uh the base two to the power of
20:14	three , it was equal to eight . That's why
20:16	we landed right there . Okay , this one is
20:20	I'm gonna save that one for later . This one
20:21	right here is over here , we can draw a
20:25	little line right here . It's two to the power
20:26	of 12 to the power of one , that's equal
20:28	to two . Now we know that this is two
20:29	to the power of two . Right ? So I
20:32	can kind of draw this one here . As over
20:34	here , this is gonna be uh to I'm sorry
20:37	3:08 . This is 3:08 , and this one is
20:42	2:04 . And this one right here is one comma
20:47	two and this is just the intercept right there .
20:50	Okay , so this one right here is to to
20:54	the power of two is four . But what I
20:55	wanna do is I want to introduce this notation .
20:57	Notice over here , I have B . To the
21:00	power of K . Is in the K . Is
21:02	the X . Moment the bases be in is the
21:04	number you get out . Uh that the exponential calculates
21:08	in . So what we really have here is I'm
21:11	gonna say this right here is uh no , it's
21:15	really the number two , but I'm gonna put the
21:16	letter K underneath it . So really this thing is
21:19	going to be something like a comma in why ?
21:22	Because two to the power of K . Was equal
21:25	to end . Yes , I know it's at the
21:27	number four , but I'm generalizing it because I want
21:29	to map it over . You'll see why I'm gonna
21:31	generalize it later . So it's really the number two
21:32	. But let's call it the letter K . Because
21:34	any of these numbers can be K . I'm gonna
21:36	run it through . It's gonna be two to the
21:37	power of K . That gives me the number back
21:39	. We know it's really the number four , but
21:41	that's pretty much what it looks like right there .
21:42	Okay . Now what we want to do if you
21:44	want to draw the inverse of this exponential function ,
21:47	the inverse remember is the function that you get when
21:51	you draw a diagonal line At 45°, , which is
21:54	this line why is equal to X . And you
21:56	flip this thing over ? Okay , now I don't
21:58	have room on this graph . So what I'm gonna
22:00	do is go over here and try to draw it
22:02	right here . So here is this gonna try to
22:05	draw as good as I can not gonna probably do
22:08	a great job . And this is not gonna be
22:10	ffx , this is gonna be f uh inverse of
22:14	X , right ? The inverse function , which means
22:17	that I have to draw kind of this diagonal line
22:19	here , which I will do is a little reference
22:23	. It's gonna be a diagonal line . And this
22:25	diagnosed line is the equation why is equal to X
22:27	. And we're gonna essentially you have to pretend it's
22:29	here and you map it and you flip it over
22:31	. Right ? So how many tick marks do I
22:33	need ? Let's try to use a similar colors .
22:35	Let's go if you think about it . This is
22:37	gonna flip down . So you need eight this direction
22:39	123456788 And then let's just do 1234 . We need
22:47	I think it's four in this direction like this .
22:49	All right . So what's gonna happen is this thing
22:52	is gonna get mapped over here ? So what's gonna
22:54	happen is let's extend this guy down just a little
22:58	bit . Remember ? The inverse function takes all of
23:02	these points and it interchanges them . This point gets
23:06	interchange . This point gets interchange . That's how the
23:08	mapping happens . So , we know that zero comma
23:12	one is here . So that means it needs to
23:14	go through one comma 01 comma zero . Right here
23:17	. Actually , let me Yeah , let's do it
23:19	right here . 1:00 needs to go through this point
23:21	, right there . One comma two is a point
23:24	. That means two comma 12 comma one . This
23:26	needs to be a point . Is that flip it
23:29	around ? Two , comma four . That means four
23:31	comma two needs to be a 20.1234 comma 12 means
23:36	this guy right here , one . What's 1234 slips
23:39	into the wrong ones right next door right here .
23:42	1 , 2 , 3 4 comment to and then
23:44	three comma eight means flip it over eight comma three
23:46	. This is eight comma three . This point needs
23:49	to be in right here . So all I do
23:51	is take these , I flip them around , they
23:53	draw the points and then you can kind of see
23:54	that it's gonna bend over like this . So if
23:56	I can do a good job , which I probably
23:58	can't , it's gonna go through like this , bend
24:00	over and go something something like this , which actually
24:04	is not too bad . Yeah , I kind of
24:06	messed up the end there . You get the idea
24:07	, there's something like this . That's not too bad
24:10	. And if you use your imagination , you can
24:11	see that this thing reflected over is the exact inverse
24:15	because it's a flipped over version of this thing .
24:17	So this equation down here that we've written down is
24:20	not to to the X . This thing is called
24:23	the inverse . Yeah . Right . And it is
24:27	log base two of the number in . Remember you're
24:33	taking the longer the using a given base of the
24:36	number in . And what does it give you back
24:38	? It gives you back the exponent . The exponent
24:41	such that that base to that exponent is that number
24:44	? You're literally going backwards . It's like the logarithms
24:46	, Like we never calculate the exponent , that's that's
24:49	never done . But logarithms do that . They give
24:51	you the exponent back , right ? They give U
24:53	K back , they give you the K . Which
24:55	is the exponent needed when you raise it to the
24:57	base to give me this number right here . Right
24:59	. So every number I put in here the law
25:02	algorithm has given me the exponent needed back . Right
25:05	? So what are these points here ? Okay this
25:08	point means that I put um This point means that
25:13	I put 1:00 in there which is exactly the mirror
25:17	image of this guy . Okay or the flipped value
25:20	of this guy . It gives me the exponent back
25:22	. In other words I'm gonna take the law algorithm
25:23	based two of the number one . The exponent that
25:27	that I need to do that as a zero because
25:28	two to the zero power gives me one . Okay
25:32	um This point right here is two common one .
25:39	Because if I'm taking the law algorithm based two of
25:42	the number two I'm going to get a one back
25:44	. I need an exponent of one to do that
25:46	. Okay this guy is I'm gonna write two things
25:50	down . It was two comma four . It's really
25:53	four comma too . However I'm gonna also write it
25:56	as instead of K . Dot comma in its in
25:59	kama . Kay because what I'm doing is I'm putting
26:02	the number in that I want to take the log
26:04	rhythm of and it's giving me the exponent back in
26:06	this case the exponent was too but in general has
26:08	given me the exponent K back and then this last
26:11	one here Is instead of 3:08 It's 8:03 . So
26:18	remember what did I say ? Expo exponential functions do
26:22	I'm sorry what what did I say inverse functions do
26:25	it under does or does the opposite of the original
26:28	function . So if I have a function here number
26:30	one the function here number two and I put the
26:32	number one in there then I'm gonna get some value
26:35	out and I stick that and put it through the
26:37	inverse . I'm gonna get some other number out .
26:39	But the number I get out will be exactly the
26:41	same as what I put in because It kind of
26:43	undies the function of the first calculation . They put
26:46	a two in and run it through both of them
26:48	. I should get a two out , put a
26:50	10 in and run it through both . I should
26:52	get a 10 out . So let's look at that
26:55	in this exponential function . If I put a three
26:57	in as an input , I'm gonna get an eight
26:59	out of this exponential function . But if I take
27:02	this eight and I put it through the second function
27:04	, I'm gonna get a three out . Three is
27:06	exactly what I put in to begin with . If
27:09	I put a two into this original function , I'm
27:11	going to get a four out . But if I
27:13	put that four in here , I'm gonna get a
27:16	two out , which is exactly what I started with
27:17	. So you see these are inverse functions because whatever
27:20	I put into the exponential function and then I run
27:22	it through the inverse , which is the algorithm with
27:25	the same base , the base has to be the
27:26	same . What I get out is exactly what I
27:29	put in . And that means of course this isn't
27:31	a mirror image reflection of this thing here . So
27:33	you need to remember two things about this graph number
27:36	one . This is a law algorithm , it starts
27:38	it gets infinitely close to the axis here , it's
27:42	an asem toe , but it goes off to infinity
27:45	this way , but it bends over very , very
27:47	quickly like this . Okay , whatever number you put
27:50	into the function , it's giving you the exponent out
27:53	needed , so that if you run it backwards through
27:56	the function , you kind of get what you started
27:57	with . And then of course it's a mirror image
28:00	reflection of this exponential function . So these guys in
28:05	verses of each other . So we don't say that
28:13	this is just the log is the inverse of the
28:16	exponential . We also go the other way we say
28:18	that the exponential is also the inverse of the law
28:20	there in verses of each other and they undo each
28:23	other . So if you have an equation that has
28:25	an exponential function in it , but that variable is
28:28	wrapped up in the exponential function , you want to
28:30	kill the exponential . You might take the law algorithm
28:32	of both sides because the law algorithm is going to
28:34	kill the exponential , it's gonna disappear because of exactly
28:37	what we said here in versus undo each other .
28:39	Right ? If you have the other situation , you
28:42	do the other operation , like if you have a
28:43	low algorithm on both sides of an equation or one
28:46	side of an equation but your ex your variables wrapped
28:49	up inside of the law algorithm . But you want
28:50	to solve for the variable . You gotta kill thatl
28:52	algorithm . So you raise both sides of the equation
28:57	to the to the exponent base to the what you
29:00	basically have to raise it to an exponent to kill
29:03	the algorithm and they'll disappear in your variable pop out
29:06	. We'll do that later . Will solve equations .
29:07	Using this property here . The bases are the same
29:11	. That's the other thing I want to point out
29:13	in order for this inverse business to happen . The
29:17	basis are the same , right ? The base two
29:22	to the power of X . The inverse of that
29:23	is the base too long rhythm . And have the
29:25	bases have to be the same . Otherwise they're not
29:27	the same thing . And that's why when we solve
29:30	logs and we do this business with solving logs ,
29:32	we end up writing it as an exponential function because
29:35	they both kind of go together like that . And
29:37	any log can be written in exponential form and then
29:40	the exponential can be written in terms of a log
29:42	form . Because what's happening in this log equation is
29:44	I'm just giving you a number and getting the exponent
29:46	out . When I go to this equation , I'm
29:48	giving you the exponents and I'm giving you the number
29:50	out . So it's literally like going backwards through the
29:53	operation there . All right , So I want to
29:56	talk about a little bit shifting the discussion to why
30:00	we care about logs . There's so many reasons I
30:02	can't give it all in one lesson , but I'm
30:03	gonna give you a couple of really big ones right
30:05	now . Um so far we've been doing a lot
30:09	of base two logarithms . The reason we're doing base
30:12	two logarithms because it's easy to calculate . But really
30:15	one of the most common logs that you're gonna run
30:17	into is base 10 logarithms . In fact , when
30:20	you say the word law algorithm and if you don't
30:22	specify a base at all , most people are going
30:24	to assume that you probably mean a base 10 logarithms
30:27	. And later on in the class , we're gonna
30:29	talk about base L algorithms . The special number E
30:32	. We'll talk about that later . We're not gonna
30:33	get into that now , that thing is called the
30:35	natural algorithm . We'll talk about that much later .
30:37	For now . Let's focus on the other special algorithm
30:40	which is the base 10 logarithms . Alright , so
30:43	base 10 logarithms are really important . So let's talk
30:45	about that . Base 10 logs . All right ,
30:51	so um what I mean when I say base 10
30:53	logarithms is is if you have a function F of
30:57	X Equals 10 to the power of x . the
31:00	basis 10 then it's inverse . Yeah . Is going
31:04	to be a long algorithm . But the law algorithm
31:06	has to be a base 10 as well algorithm .
31:09	Base 10 of some variable X . Okay . So
31:13	you see in order for these to be in verses
31:15	, the bases have to be the same across the
31:17	exponential and across the algorithm . Otherwise they're not in
31:19	verses of each other . So let's talk a little
31:21	bit about Some base 10 logs . Let's say I
31:25	have log of the number one notice I didn't write
31:30	the base 10 . If you see a log without
31:33	any base written at all , you just pretty much
31:35	assume it's a it's a base 10 law . That's
31:36	how common it is . If you see a base
31:38	there you have to use the base . But if
31:40	you don't use a base at all , if you
31:41	don't see a base at all , it's a base
31:43	10 log . What is the log base 10 of
31:45	the number one ? Well , what we know is
31:48	that that means that's exactly the same thing . Is
31:50	this base 10 logarithms of one . And what this
31:53	means is we take the base of 10 , raise
31:56	it to some unknown power . We're trying to calculate
31:58	because the law gives you the power back Equals one
32:01	. So my question to you is what value of
32:03	X . works ? And that means X . Has
32:06	to be zero . Why ? Because tens of the
32:08	zero is one , Right ? So what we've learned
32:11	here is that log of the number one ? Base
32:13	10 log of the number one is zero . Okay
32:16	. What I'm gonna do is calculate a few of
32:18	these things going down the page and I'm gonna draw
32:20	some really important um observations as we go down here
32:25	. So that was a log of the number one
32:26	. Let's take a look at log of the number
32:29	10 . Again , it's a base 10 law because
32:31	there's nothing written there , but you can kind of
32:33	assume that it is , this is the exact same
32:35	thing as writing base 10 logarithms of the number 10
32:39	. So what does this mean ? It means the
32:41	base so the power of something has to equal this
32:44	number . Now . What does this experiment have to
32:46	be equal to ? It has to be equal to
32:48	one , The only exponent that works as one .
32:51	So that means that the log of the number 10
32:54	is equal to one . So we figured out the
32:56	log of 10 base 10 and the log of 10
32:59	is one again based 10 . So let's do a
33:02	couple more examples . Uh Down the page here ,
33:06	Let's go and do the log of 100 . Okay
33:10	, how do you figure this out ? Well ,
33:12	you know , it's a base 10 by now ,
33:13	so 10 to the power of something is 100 .
33:16	What does this exponent have to be ? The only
33:19	way this works is if x is two because 10
33:21	squared is 100 . So we've learned that log of
33:24	100 is equal to two . So I'm I'm generating
33:27	a pattern here and now that we understand what we're
33:30	doing , we're gonna go down the page a little
33:31	bit faster if the log of one is zero and
33:34	the log of 10 is one and the log of
33:36	100 as to what do you think That the log
33:38	of 1000 will equal to ? Well , it's gonna
33:40	be 10 to the power of something as 1000 .
33:42	So it has to equal three . Right , What
33:45	do you think the log of 10,000 Is going to
33:50	be equal to 10 to the power of something as
33:51	10,000 has to be equal to four . You see
33:53	what's happening , you started out at zero and then
33:56	it goes one , then it goes to , then
33:58	it goes three , then it goes four . So
33:59	as I take the law algorithm , I've multiplied by
34:02	10 . I'm taking the logarithms , something 10 times
34:04	bigger . And then here I'm taking the log rhythm
34:06	of 10 times bigger still . And here I'm taking
34:09	the longer than the 10 times bigger . Still .
34:10	10 times bigger . Still , every time I go
34:12	up times 10 , the longer than just goes up
34:16	by one . You see I'm taking the long rhythm
34:18	of something 10 times bigger every time . But the
34:20	longer term only goes up from 0 to 1 to
34:22	2 to 3 to four . And you can generalize
34:25	that . What if you do something that's not a
34:26	perfect little times 10 thing . Let's take the log
34:30	rhythm Of 20,000 . That is not 10 times bigger
34:35	. It's not 10 times bigger . That's only two
34:36	times bigger . What do you think you would get
34:39	? What you're going to have is you're gonna have
34:40	10 To the power of something is 20 1000 .
34:46	When you run that in a calculator and figure out
34:49	what this exponent is . You're gonna get 43 Now
34:53	I've rounded it to two decimal places . But you
34:55	see it's just a little bit bigger than this one
34:57	. See the algorithm of 10 , was four ?
35:00	The algorithm of 20,000 was just a little bit bigger
35:02	than four . Okay . What do you think the
35:04	logger them of ? 150 would be just to pick
35:08	a totally different number ? Well , what you would
35:10	say is 10 base 10 to the power of something
35:13	is 1 50 . When you run that through the
35:15	calculator , you're going to get 2.18 And I'm getting
35:20	to a point here . I promise I have one
35:21	more to put on here . What about the law
35:23	algorithm of 1700 ? When you run that to a
35:28	calculator , you're gonna get 3.23 . Okay . What
35:33	I'm trying to get you to say is something that
35:36	I honestly didn't really realize about logarithms until Well ,
35:40	well beyond I had learned them . It's something way
35:43	, way , way in the future . I finally
35:45	understood that logarithms really , I don't wanna say they're
35:47	only use but one of their main uses . Okay
35:49	, when you run a number through algorithm , what
35:52	the law algorithm really is doing is it's giving you
35:54	the exponent back . That's what I've been telling you
35:56	over and over again . But notice that for a
35:58	1000 what's happening is the number you're getting back is
36:02	the number of decimal places past the first position .
36:04	When you take the log base 10 of 1000 you're
36:07	getting the three , it's telling you , hey there's
36:09	three digits past the number one . When you take
36:12	the log of 10,000 and get a four back it's
36:14	telling you , hey there's four digits past the one
36:17	when you take a log of 100 is telling you
36:19	, hey there's two digits past the one . Hey
36:21	there's only one digit pass the one . Hey there's
36:23	no digits past the one . So the longer than
36:25	base 10 is really telling you how big the number
36:28	is , how many digits it is , It's ignoring
36:30	all the details . Notice that the log of 20,000
36:34	is telling you , Hey there's just 4.3 digits .
36:37	That's kind of a little weird because there's four digits
36:40	after here . But it's basically telling you how close
36:42	you're gonna be because when it gets to five that
36:44	would be five digits past , right ? It's at
36:47	20,000 you have to go 20 and 40 and 60
36:49	and 80 . Then you will get to 100,000 .
36:51	That would be five digits past . So this is
36:53	telling you you're getting fractionally a little bit closer to
36:56	that next decimal . Place that next zero in the
36:59	number . It's telling you how much closer you are
37:02	to that but it's basically reporting back how many zeros
37:06	or how many digits you have in the number past
37:08	the first position here , you had none past the
37:11	first position here . You had one past the first
37:13	position to pass the first position . Three past the
37:15	first position four passed . This is just a little
37:18	bit more past four . This is a little bit
37:20	more past two so it's 2.1 a . This is
37:23	a little bit more past three . Getting close to
37:25	you know you have to get up to 10,000 to
37:28	get to the next one and you're getting a little
37:30	bit closer and the decimal part of it is telling
37:32	you that . So it tells you the number of
37:34	digits . So if you have numbers where you don't
37:37	care about the exact value of the number , you
37:39	just kind of want to know roughly how big things
37:41	are . Then the logarithms a perfect thing because it's
37:44	gonna throw away all of the , I don't want
37:46	to throw it away but it's going to tell you
37:48	basically how big the number is , how many digits
37:50	the number has Without all the details of that ,
37:53	of the actual number there you might say . Why
37:56	do we care about that ? Why don't we just
37:58	give you the number ? Well , a great example
38:00	is the Richter scale of of of uh of earthquakes
38:05	you always hear in the news 6.2 on the Richter
38:07	scale , 5.3 on the Richter scale . And it
38:10	also doesn't really become very clear what those numbers really
38:13	mean . But you have to remember that every time
38:16	we multiplied by 10 of the number , the log
38:19	rhythm of it just went up by one . So
38:21	what we've learned here is since the Richter scale is
38:23	log arrhythmic . If you see a four point oh
38:26	on the Richter scale and a 5.2 on the Richter
38:29	scale , it doesn't seem like very much , but
38:31	really that means it's 10 times bigger . A four
38:34	on the Richter scale and a five on the Richter
38:36	scale is not a little bit different . It's 10
38:38	times bigger . How do I know ? Because the
38:41	difference between three and four or let's go back to
38:43	three and three and four versus four and five ,
38:45	The difference between three and four on the Richter scale
38:47	is actually 10 times bigger in energy , earthquake is
38:50	measured in energy . Right ? So if you have
38:54	a situation in an earthquake , right ? I'm not
38:57	an earthquake scientist . Okay , But here , you're
38:59	basically measuring energy energy of the earth coming coming out
39:04	in terms of , you know , um in terms
39:08	of uh this guy here , let's let's talk about
39:11	versus time . So energy versus time , an earthquake
39:14	probably starts small and it is really , really powerful
39:16	and it gets really small again . Right ? So
39:18	what you have with earthquakes is there is a huge
39:22	amount of difference in energy released . Right ? So
39:25	what might happen is you might start out really ,
39:27	really , really low in this energy scale might be
39:30	really , you can't even read it on this graph
39:32	and you have a little spike here , a little
39:34	bit bigger spike . And then the main part of
39:36	the earthquake comes , it is really , really big
39:38	. Really , really big and has jumped down here
39:41	. You can't even read and it's really , really
39:43	big again , Really , really big again . And
39:45	then it gets off the chart and it comes back
39:48	down like this . You see a graph like this
39:50	isn't that useful to us . I mean it is
39:52	useful , it does tell you how big everything is
39:54	. But the problem is I can kind of read
39:56	these but I can't read any of this stuff down
39:58	here because at this scale the tiny little wiggles at
40:02	the bottom are impossible to read because they're just so
40:04	small compared to everything else . And the reason they're
40:07	so small is because there's a huge difference in energy
40:10	released in the beginning of the quake to the middle
40:12	of the quake . Right ? So what you need
40:14	is a way to tell me how big these different
40:17	parts of the earthquake are or how big different earthquakes
40:20	are without using the actual numbers . The actual energy
40:24	numbers . Because the energy numbers are going to be
40:26	huge , 100 million , trillion . And then another
40:28	earthquake might be 100,000 or 5000 . And that number
40:32	is just so different that you can't graph it like
40:34	that . So what we do is instead of reporting
40:36	the energy numbers of the earthquake , we take the
40:38	algorithm Right now , the actual Richter scale is a
40:41	little more complicated than this , but basically you're taking
40:44	the law algorithm . So notice that when we multiplied
40:47	like this , this the difference here in energy here
40:50	to the energy here might be a difference of 100
40:52	or a difference of 1000 . It's hard to graph
40:55	that . But the difference between 100 and 10,000 is
40:59	just a difference of two points on a long algorithm
41:01	scale . So actually we graph these earthquakes on what
41:04	we call la algorithm scales . We think the law
41:06	algorithm of all the data and that basically gets rid
41:09	of all the details and it just tells me how
41:10	many digits because the numbers are so huge . I
41:13	only care about how many digits are in there when
41:15	it's little versus when it's big . So the actual
41:18	Richter scale looks something like this . So it's called
41:21	Richter scale . And again , I'm not an earthquake
41:26	scientists , I grabbed these right off Wikipedia . Right
41:29	, so a one on the Richter scale is called
41:31	a micro . That's , you can basically can't even
41:34	feel that A two on the Richter scale is called
41:36	a minor earthquake . Three is also characterized as minor
41:42	. You probably feel that , but you're not gonna
41:44	be that scared by it . A four is going
41:46	to be called a light light , you're gonna have
41:49	some damage , but probably not very much . A
41:52	five is going to be moderate . A five is
41:56	when you're gonna start reporting these things on the news
41:59	. A six on the Richter scale is a strong
42:03	A seven on the Richter scale is a major right
42:09	? An eight on the Richter scale is great .
42:13	A nine on the Richter scale is total devastation .
42:22	Right ? A nine point anything . Now there's no
42:23	10 because it only goes to 9.99 or whatever .
42:26	Okay . But basically anything above a nine , your
42:29	city is flattened . Major major damage . I mean
42:31	, way more major than a typical earthquake . I
42:33	mean , the reason that these numbers don't often convey
42:36	the strength here is that because the number two and
42:38	number four , they don't seem so far apart .
42:40	However , between the # one and 2 , This
42:42	is times 10 bigger between two and three . This
42:45	is times bigger than that again . So that means
42:47	between one and three is actually 100 times bigger Because
42:50	10 times 10 , right ? This is times 10
42:54	. This is times 10 . And then from here
42:57	to here is times 10 and then you have times
43:00	10 , you get the whole idea times 10 Times
43:04	10 . So if you're looking at the difference between
43:06	one and two , that's 100 times bigger . Between
43:09	one and three , I'm sorry , one and uh
43:13	this one right here . This is yeah , this
43:15	is 100 . This would be 1000 times bigger .
43:17	This would be 10,000 times bigger . Right ? And
43:20	you can go down the calculation and figure out how
43:21	much bigger A nine would be than a one .
43:23	But that's why when you see on the news ,
43:25	you might see a 5.3 on the Richter scale and
43:28	that's bad . But a 6.3 is way way worse
43:32	because it's 10 times bigger . Okay . And that
43:35	is one of the biggest uses of the of the
43:37	law algorithm is in practical use that you would see
43:40	in everyday life . But there's many other examples I
43:42	can give you from chemistry , I can give you
43:43	from physics or whatever , but this is the one
43:45	that you'll actually probably see on TV . And it
43:47	all goes back to the fact that when you start
43:50	taking logarithms of numbers , if you on a base
43:52	10 Then you start multiplying by 10 , then the
43:55	actual algorithms are just giving you the number of digits
43:58	back . So they're they're only going up by one
44:00	each time every time you multiply by 10 . So
44:03	that is the concept of what is a law algorithm
44:05	. I hope that you can understand what algorithm is
44:08	. It's basically the opposite . Also what we call
44:11	the inverse of an exponential function law algorithms are not
44:14	going away . Some students don't like them but you're
44:16	just gonna have to get used to them . The
44:18	best advice is just to when you start to see
44:20	a longer than written down anywhere . Just immediately convert
44:23	it to exponential form . Because most people are more
44:25	familiar and comfortable with an exponential function log rhythms get
44:30	, you know crazy and people get crazy . I
44:32	don't understand how to deal with it , but we
44:33	have to learn it because later on we have laws
44:36	of logarithms , how to ADL algorithms , how to
44:38	multiply logarithms are not going away . So make sure
44:41	you understand this , solve these problems . Draw these
44:43	graphs , follow me on to the next lesson ,
44:45	we're gonna start simplifying expressions that have logarithms and we'll
44:48	be using this definition , this exponent inverse definition along
44:52	the way . So follow me on there will continue
44:54	learning about logarithms in math .

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