Logarithm Change of Base Formula & Solving Log Equations - Part 1 - [7] - Free Educational videos for Students in K-12

Logarithm Change of Base Formula & Solving Log Equations - Part 1 - [7] - By Math and Science

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00:01	Hello . Welcome back . I'm Jason with math and
00:02	science dot com . Today . The title of this
00:04	lesson is called la algorithm change of based formula and
00:08	also solving more exponential and log rhythm equations . I
00:12	told you at the very beginning the most important thing
00:14	about logarithms that we're going to use them for at
00:16	least in the near term is in solving equations and
00:19	specifically solving exponential equations . And we have solved some
00:23	exponential equations and we have used the algorithms but by
00:26	now you probably figured out that some equations become difficult
00:30	to solve if the law algorithms or to get to
00:33	get a numerical answer for if the law algorithms that
00:35	you're dealing with are in crazy weird bases . Like
00:39	for instance we've we've talked about the fact that you
00:41	can have logarithms in a base to you can have
00:44	based five logarithms , base three logarithms . So you
00:47	have those exact answers if you can take logarithms with
00:49	all these different bases . But a lot of times
00:51	what happens is you have a calculator in your hand
00:54	and you want a number . But the calculator doesn't
00:56	have usually a law algorithm With the base five and
01:00	a six and a base 10 and 11 . They
01:01	usually don't have all of those things . So how
01:03	do we evaluate and solve equations to give numerical answers
01:07	? When the only button on the calculator we have
01:09	is the regular law algorithm button ? How do we
01:11	do that ? So it turns out that there is
01:14	what we call a log a rhythm change of base
01:17	formula . And that is a really neat little tool
01:20	to have in your back pocket because it allows you
01:22	to take logarithms of any base you want by changing
01:26	the way that the problem looks . And it's a
01:27	little bit complicated to say in words . So I'm
01:29	going to write it down for you before we get
01:31	to the actual change of based formula . I want
01:33	you to kind of just pull your calculator out whatever
01:36	little calculator you have generally there's gonna be too low
01:39	algorithm buttons on the calculator . Okay . Um and
01:43	uh the first one will be called L . O
01:45	. G . If you just see an L .
01:47	O . G . A log button , it means
01:48	it's a base 10 logarithms . You don't have to
01:51	put the number 10 there Little based in there because
01:54	you know , if it's his blog , it's base
01:55	10 . So L . L . G . On
01:57	your calculator means base 10 . Right ? And then
02:02	you have another button that's called L . N .
02:04	And this thing is called base . It's called a
02:06	natural algorithm . And this is the base mm uh
02:12	longer . And we're gonna study based e natural algorithms
02:14	later . I don't want to get into it now
02:16	. But just suffice to say there's a very special
02:19	base uh with labeled with a letter E . And
02:23	that is just a number . E . Is a
02:24	rational number like pi or like squared of two E
02:27	. Is 2.71 and then infinite , non repeating decimals
02:32	after it . We're gonna get all into Y .
02:33	E . Is important a little bit later . So
02:35	forget about it for now . But I want to
02:37	just let you know that your calculator is only going
02:39	to have two buttons , Base 10 logarithms and base
02:42	E . L . Algorithm right now computer if you
02:45	go to a computer you can take any basil algorithm
02:47	you want . But on a handheld calculator these are
02:48	the only ones you'll probably see . So how do
02:51	we do something for instance ? Like if I want
02:54	to figure out , You know what is the law
02:56	algorithm ? Base four of the number 17 . How
02:59	do I do that ? Like how do I even
03:01	type it into the calculator ? There is no way
03:03	to do . Usually law algorithms of different bases in
03:06	a calculator . So we have a formula that I'm
03:09	going to write on the board . That's in all
03:12	of the , you know , algebra books . It's
03:13	also in the calculus books because we use it quite
03:15	a bit in calculus . Um and it's called the
03:18	change of based formula . So let me write it
03:20	down and explain how it basically works . All right
03:24	. So this is the meat potatoes of this lesson
03:26	. This is called the log change of base formula
03:36	. And you're gonna find out that it allows us
03:38	to do things like this . Uh Actually really ,
03:42	really easily . All right . And here's what it
03:44	is . It's not very hard or lengthy to write
03:46	down . Here's what it is . Large rhythm base
03:50	A . Of the number X . Any base I
03:53	want can be written as the law algorithm of any
03:57	other base I want of the number X . Same
03:59	as this , divided by law algorithm . Same base
04:03	B of A . Now I'm fully aware that most
04:06	of you that have never seen this before , we're
04:08	going to look at this and say this makes no
04:09	sense to me . So here in just a minute
04:11	it will make a lot of sense , I promise
04:13	you right . But what it allows you to do
04:16	is it allows you to take logarithms of crazy weird
04:19	bases only by using well it's used for more than
04:22	this , but mostly you're using it to be able
04:24	to do it on a calculator because you don't have
04:26	base 15 logarithms or based seven law algorithm buttons on
04:30	your calculator . I am not going to prove this
04:33	right now here in the beginning of the lesson .
04:35	What I want to do is show you how to
04:37	use it . And then if you stick with me
04:39	to the very end of the lesson , I'm gonna
04:40	drive and show you where exactly this thing comes from
04:43	. So just stick around if you want to know
04:45	where it comes from now , what this is saying
04:48	in a nutshell , it's easier to see with a
04:50	simple little example . What if I wanted to take
04:54	the law algorithm right ? The base three large rhythm
04:57	of the number nine . Now I just told you
04:59	that the base three large rhythm button is not in
05:01	your calculator . So there's no way to put it
05:02	in there . So you can see log base A
05:05	. Of the number X . This is log in
05:07	this case based A . Is the number three and
05:09	the number I'm taking the log is here . So
05:11	this is basically this what this is saying is I
05:14	can write this log rhythm as follows . I can
05:19	change the base of this log from base A .
05:21	To any other base . I want notice . B
05:23	. Is the base on the right and A .
05:25	Is the base on the left . So if I
05:26	want to I can change this thing to a longer
05:29	the base too . Of the number nine divided by
05:33	log rhythm . Base two of the number three .
05:36	I want to let that sink in for a second
05:39	. What it's saying is I'm trying to take base
05:41	three log of the number nine . I can instead
05:44	change that to the law algorithm of the number nine
05:47	. What you're taking originally you're taking the log of
05:49	over here . You're taking the log over here .
05:51	I want to make it a base to for whatever
05:52	reason , so I have to take base to algorithm
05:55	of nine . But then I have to divide it
05:56	by the base to log of what the original bases
05:59	see the basis three right here . Okay now this
06:02	isn't so helpful because I don't have a base to
06:06	log rhythm on my calculator either . I don't have
06:08	a base to button on my calculator . But mathematically
06:11	a base to log of nine divided by a base
06:14	to log of three is equal to this right here
06:17	. Now you can you can make any base up
06:19	on the right hand side that you want . For
06:21	instance , it doesn't have to be based too .
06:24	I can make this log Based seven of the #
06:27	nine . As long as I divided by log day
06:30	seven of the number three . You see what I've
06:31	done ? I'm still taking the log of nine .
06:33	I'm still taking the log of the base . But
06:35	now I've changed it to a log base seven .
06:38	Now , the real reason that we use this thing
06:40	so much is for the following . All right .
06:43	Because I can make it log base 10 of the
06:47	number nine . Log base 10 of the number three
06:51	. So you see what I can . In other
06:53	words , I can transform this log into a division
06:56	of two . Other logs have any base that I
06:59	want . I just have to take the log of
07:01	this thing and the log of this thing if I
07:03	want . Based 17 . No problem . I can
07:05	make it log base 17 of nine , divided by
07:08	log base 17 of three . I want log base
07:11	24 . I can make a log base 24 of
07:13	nine , divided by log base 24 3 . Of
07:15	course , I don't have buttons for all those uh
07:18	other weird basis , but mathematically they're correct . Now
07:21	, why do I care about base 10 so much
07:23	? Right . Because basically this means this is the
07:26	same as log of nine divided by log of three
07:30	. There is an implied based 10 here because when
07:33	you don't write it you just assume it's a base
07:35	10 . Now I do have a log rhythm based
07:37	10 button on my calculator . In fact that's why
07:40	we don't usually on hand held calculators have crazy weird
07:44	based buttons or menus or anything to be able to
07:46	take logs of different bases because you don't need to
07:48	If you need to take a crazy log with a
07:50	crazy base , just transform it to be a log
07:53	. Base 10 of this divided by a log base
07:55	10 of this . So the way you want to
07:56	read this this transformation of bases . This blog can
08:00	be written as log of this divided by log of
08:02	this in whatever base that I want . Usually you're
08:06	just going to convert it to base 10 in order
08:07	to do the calculations . Okay , let me show
08:10	you a practical example of why you might want to
08:12	do that . Let's say you have log base uh
08:17	Let's make it base four of the number seven .
08:19	And I want you to to simplify that . Tell
08:21	me what that's equal to . Well , you all
08:24	know that if I wanted to I can use the
08:28	definition of law algorithm to try and solve this .
08:30	I can say base to some number equal X on
08:33	the right hand side of the equal sign is equal
08:35	to seven . This is what we've been doing for
08:36	the definition of all algorithms . Four to the power
08:39	, something is equal to seven . That's typically how
08:42	I've taught you how to solve algorithms , you write
08:45	it as an exponential and then you you try to
08:49	figure out what this experiment is . However four and
08:52	seven have completely different bases . I cannot write them
08:55	on the left and the right hand side as the
08:57	same base . I can't do it . So there's
08:59	no way that I can solve that thing right ?
09:02	However , let's just abandon that and let's go down
09:05	here and say okay instead of evaluating it by using
09:09	the definition of the log rhythm , let's just change
09:11	the base right ? What this means is that instead
09:14	of log base for log of the number seven ,
09:18	I can write this as the log of this divided
09:20	by the log of this in any base that I
09:22	want . So I can just make it a base
09:24	10 log of the number seven divided by the base
09:27	10 log of the number four I have buttons for
09:30	these are my calculator . Now of course I could
09:32	make this a base nine log of seven and divide
09:35	by base nine log of four . Or I can
09:37	make this a base to log divide babies too long
09:41	. All of them are correct . You can pick
09:43	anything you want , but base 10 is the only
09:45	thing I have a button for So on my calculator
09:47	. I'm gonna go crank in and put log seven
09:49	and I'll get a decimal bag 0.8 451 . This
09:53	is rounded and

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