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

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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