 So looking at number two what's my base? What's my base? So I would go okay, this is the inverse of three to the x which is zero comma one one three two nine Which means the points that I'm gonna use are the inverse of this it's gonna be one zero three one Nine two oh and this has an asymptote of y equals zero this has an asymptote of x equals zero So far so good. Okay Now I'm gonna do my list. I really didn't leave you enough room here to list things carefully I would admit I didn't type this again. I'd make it bigger with more room anyhow expansions compressions well, I see a Vertical expansion by two Yeah, I See a one-half now. That's a horizontal Expansion by two because it is next to the x and everything's backwards Any reflections that's convenient no negatives one right and One down. Okay Let's walk through the transformations vertical expansion by two means the y's are going to change sure One left Absolutely, it'd be one left. That's what I wrote. I'll make that L a little clearer so you can read it better Note to self fix the recording on the internet so people online. Don't think you're stupid Note to self. Don't say those things out loud when you're recording because you can't go Vertical expansion times by two times lost my projector froze really Let's do that All sorts of funny stuff happening today Let's do that should be back again shortly have patience children. Okay, we're back one two three Okay Vertical expansion times by two times by two. Oh, that's an x not vertical Horizontal expansion times by two times by two. That's not gonna fit one left That'll become a one that'll become a five ah That'll become a negative one One left. Oh, I should also have horizontally expanded that by two, but what's two times you're still zero One down to become a negative one a one and a three. Okay Here's what I got. I got an asymptote at x equals negative one Yes, I got One comma negative one I think and I got one two three four five comma one What does every exponential graph look like like this or like this? You know what every single log graph looks like it has a vertical asymptote or I got a Vertical can't really do this one really well a vertical asymptote this log graph. I'm pretty sure just it's gonna have to curve this It's gonna go Well except mr. Do it why don't you go through the points shut up? It's harder than it looks on this tablet It's gonna look like that What's the domain Everything to the right of negative one range all reels Asymptote x equals negative one X intercept is a decimal. I wrote this up and then I went I should have not included x intercepts y intercept is Also a decimal I think So the only way to find them would be to cheat and use your graphing calculator or solve it I'm not gonna freak out about that. I'm working you do the graph right now. Is that all right? Am I right anybody get that? Yeah, I'm nodding. No. Yes. I think I'm right. Is that okay? By the way, if you want more practice, of course, you can always do The Examples that I skipped over so where it says example one two three and four if you want a bit more practice You can try those ones and check on your graphing calculator if you have a problem. Yeah Yeah, anyone else? Okay, so I have for you a take home quiz Turn in your workbooks if you would be so kind please to turn to page a hundred and fifty one page a hundred and fifty one and I apparently need to delete the ink in this lesson delete ink page a hundred and fifty one page one five one Incas vanished. Yay Now I want to preface this the first part of this lesson Tyler Ain't gonna be on your test The second half yes And I'm gonna tell you though this concept the title says Identities the concept of an identities is going to be a major part of trig a huge part of trig a phenomenally large part of trig, so I'm showing you the concept of an identity 151 I'm showing you the concept of an identity right now to let it actually kind of percolate so identities Says in mathematics. It's important to understand the difference between an equation And an identity This is an equation and the reason it's an equation is only a few values of x satisfy it You could solve it as our the values are negative two and positive two This is an identity There is an equal sign. There is an equal sign. There is stuff on either side There is stuff on either side, but this is true for an infinite number of x's in fact for any valid Value of x you put it in this side you put it in this side You'll get the same answer and identity is essentially a true statement not an equation a statement That's true for not just multiple values, but all values of a variable x doesn't seem very profound yet but Let me tell you another identity don't write this down. We didn't call it an identity. We called it a rule that was the power rule is That true for any value of x that you're allowed to stick in there any x that's positive and any base that's positive but not equal to yes So that's an identity One of the things we're going to look at in trig is Proving identities. It's probably going to be the single toughest concept of the entire year But we're going to spend like I think six lessons like we're going to do it very patiently very carefully We've already met the following identities the base change law The log base B of C is the log base A of C over the log base A of B The product law multiplying two logs inside the log is the same as adding outside the law The quotient law and the power law We're going to look at today a little bit is trying to prove them Now here's the problem Even though there is an equal sign Because we're trying to prove that they're equal Cara We can't use our equation solving tricks We can't add five to both sides for example because that would be assuming they were equal in order to prove They were equal we can't divide both sides by two because that would be assuming they were equal in order to prove They were equal trying to prove that an identity is equal They are the same thing you have to only work on one side or Only work on the other side, but you can't cross stuff over otherwise they'd be Easy, but that'd be because you were assuming they were equal to prove that they were here's an example example one Says use the following steps to prove That this works we looked at the pattern a few days ago. We say that seems to work It's how we did the log base five of 17 We said that's ln 17 over ln 5 or log 17 over log 5 or log base whatever of 17 over log base same Whatever of 5 Prove it. Okay We would let if we were going to do this one the log base B of C B X There's our first step Spencer, what's the second step? Can you rewrite that as an exponent, please I yelled at you guys last class in love saying it's high time that you were able To glance at a log and see the exponent so Spencer my friend. What to the power of what equals what? No, you can't say log. We're changing it to exponential form. What to the power of what equals what no more logs? Darn right with authority though. No more questions. Absolutely mister do it By the way, this is going to be your first question on your test I'm either going to give you a log and say write it as an exponent I'm going to give you an exponent say write it as a log and I would consider that Less than C minus level question Like this is the one I expect it. Well, this is the one if you get that one wrong you're flunking the test B to the X equals C What does it say for step two spence? Okay, so I'm going to go like this log base a of B to the X equals log base a of C Now what use the power law. Where could I use the power law? Ah What can I do with that X and then it says solve for X? Get the X by itself Brett. How would I get the X by itself? Absolutely, so I wrong Brett that Brett X equals the log base a of C all over the log base a of B and Then for the final step, and I'm kind of running out of room here I had to write a little large replace the X with the original X value, which you said was log base B of C right You see what you have you have the log base B of C that was your original X value But you've also shown that it can be rewritten as the log base a of C Over the log base a of B and what you've just done Alex is you prove the power the change base change Now that's a fairly complicated proof. I'm not going to be asking you to do one like that What I will be asking you to do when we get to trig is what are called t-table proofs Called t-tables because the table looks like a capital letter t turn the page Example three is really what I want to jump down to says this Prove That this is the same as this prove it and What we always do is we draw a great big capital letter t we call this side left side or sometimes even LS abbreviation for left side we call this one right side or sometimes you abbreviate it as what? Qz what are you done? Yeah, RS? Okay, so I'm gonna first thing I always do if they haven't done it for me already I write down the question because it's no longer blank, and I feel better, so I'm going to write down log x cubed plus log of 1 over x equals to log x But not equals I've got a line down the middle that line down the middle is saying you haven't proved they're equal yet Don't you dare cross things over? Don't you dare add five to both like for example? You might think Kara could I minus this to that side? No, because that would be assuming they were equal in order to prove they're equal even if that made the question way way way way easier No, whatever. I'm doing an identity. My rule of thumb is very simple. I start with the uglier side Which side looks yuckier? Left side I'm gonna start with that It's always easier to take something yucky in math and tidy it up But to take something tidy and somehow break it down and make it yuckier, so I'm going to well What could I do with this? left-hand side Make it one log. How did you figure that out? How many logs are there on the right-hand side? One why don't I try right in the left-hand side as one log if I'm trying to show they're equal? So one of the things I'll teach you is whenever you're doing an identity You always flip your eyes from side to side side to side side to side trying to say well There's oh there's hints in what the other side looks like can I combine these are my bases the same? Yes Then let's combine these This is going to be positives on top oh I think it's going to be that actually doesn't it simplify to that log of it's really x cubed times 1 over x because it means multiplied, but wouldn't that be x cubed times 1 divided by x? Yes, oh Does that simplify how many x is on top? How many on the bottom how many left and where what are we trying to show Ellen? Now You will find especially in trig when we're doing like eight line identities that are long and tough Two or three lines from the end often you'll see it You may even get a little nerdy adrenaline rush when you see it seriously So I don't know if you suddenly saw it here, but right when I wrote that down I was oh wait a minute Ellen. What can I do with this too? Does the left-hand side equal the right-hand side? Yes Some teachers will teach you to do this. Don't write this down. What do you think LHS equals RHS stands for? And you can Justin am I somewhat of a nerd just a tad Traditionally for actually for hundreds of years after a proof Back when everybody spoke Latin you would write you can write that down if you want to you don't have to but you can It's Latin it's there for maybe here speak Latin and know what it stands for Had a German student a couple years ago who we did Stands for quo Erthaus demonstrandum Which means which was to be demonstrating which I was trying to prove We used to joke in rock Ryan in university that stood for question easily done And was the biggest lie on earth because most of my university proofs were four and five days Like 40 lines. Oh, there are proofs that are several thousand Fermi's last theorem which was finally proved about 15 years ago by Andrew Wiles I believe this proof is about 600 pages long and probably only about a hundred people on the planet can follow it They checked it and it's correct So three lines nothing. I always like to write QED. That's that's you know when they score a touchdown They spike the ball. That's me spiking the ball You don't have to but it's also nerdy. I like pull out the lap example for example for Prove this Okay, which by the way normally I would write down the question But I've done this one before and I know we're gonna run out of room So just this one if you did that's fine, but there's just this one time. I'm not gonna because it's typed up there Which sides uglier I think definitely left side, okay What else do you notice looking at the left side and looking at the right side because there is something here that's giving me a real hint as To what my strategy is gonna be Do you see something on the left side that doesn't match the right side the bases? I'm positive. I'm gonna use the base change law What am I gonna change the bases to I think 10 because first of all that's one that we kind of like That's the one on the right-hand side and what are the two bases on the left-hand side? 10 See I was able to kind of put those kids five and two come on ten So what's the base change law? Okay? one over plus one over if I wanted to rewrite this as base ten I would write log base ten of X over log base ten of five Mr.. Dook you didn't write the tens when I don't write a base Balmoneek What is the base automatically assumed to be ten? It's a three-level fraction shut up and deal with it and This is gonna be log base ten of X over log base ten of two I have not yet got my nerdy adrenaline rush I have no idea where this is gonna end up. I would probably do this just to tell myself That was one term and that was one term. In fact, I think what I'm doing here Justin is I'm dividing by a fraction Multiplying fractions is the easiest top times top bottom times bottom How do you divide by a fraction? Oh? Someone remembers their math eight if I had to go three divided by five sevenths. I don't what do I do instead Brett? three times Seven over five if I have to go one divided by this I don't I'm gonna go one times this flip This is gonna be the same as one times the log five on top over the log of X plus one times The log two on top over the log of X By the way, Tyler will this one times and this one times make any difference at all In fact, I wouldn't have freaked out if you had just said, hey, I Know that simplifies to that I'm gonna have to do this and I'm gonna have to do this I'm gonna run out of a room otherwise So now what how do I add since how many fractions do I have on the left-hand side here? To how many do I have on the right-hand side? Let's see if I can combine these as one fraction. Now, how do I add fractions? Hmm? How do I add fractions? What do I need before I can add fractions? I need a common denominator. Oh Wait a minute. What's my denominator in the first fraction? Log X. What's my denominator in the second fraction log? I have a common denominator. I Can add this now if you suck at fractions up here. You can make a little note Two over five Plus one over five. It's not three over ten. What's two over five plus one over five? Three over five you add the tops and keep the denominator when you're adding fractions so let's add the tops and Keep the denominator Now I have no idea where I'm going But I'm just being meticulous careful and trying to be a bit clever by looking at the right side and saying what does it look like? Now what? Hmm. Oh Did you get your nerdy adrenaline rush? Did you see it already where we're gonna end up? Because you're saying that these bases are the same adding is the same as what and I'll get the log of what ten What is the log base ten of ten? What do I have on the top on the right-hand side? Oh I think we can do this This is the log of five times two Over the log of X and I'm gonna go cross out the five times two. I'm gonna put a ten there and what's the log base ten of ten What's the log base ten of ten? Turns out it works out to a one over and on the bottom I have a log it is base ten what I don't write a base of X coal air-toss demonstration Which was to be demonstrated question easily done. I kicked it Justin. These are weird I'm fully expecting you to have a tough time with these if you do your normal If you don't you're one of the Eric Bing freak types, which is good Helpful, but what I want you to see like if you find this tough. I don't mean like oh, I'm stupid No, these are done I'm gonna give you a few to try and then we're gonna go to part two of the lesson See if you can prove number three Number three shouldn't be too bad. See if you can prove number four Take a look at number four. Can you tell me what strategy you're probably going to try? There's one that's screaming out to me as the first thing. I'm gonna try base change. You know why three different bases Probably try changing everything to base C. Hey, I don't know try number five Five a is pretty tough eight is good Nine is good Then what I would like you to do No, can you turn me pause for a second in the video here So if you want to turn to the blank page page 150 and Write a little heading. We're gonna practice some if then questions because there's a couple on your quiz So as a heading on page 150 you can write if then questions This is that weird one that I made a bonus question on the last quiz and now I said hey no longer bonus fair game These are almost always multiple choice on your test on the provincial exam They were always multiple choice and since this is the last year. I'm teaching math 12 I'm not gonna change your tests. So here is your first one It says if log bait well copy it out if log base three of y equals c minus log base three of x Where y is positive and x is positive because you can't take the larger than negative Then y is equal to See a few of you squinting you do that It's about as big as I can get it if you're still squinting Two words black says and if you want to also on the left-hand side Quickly copy out the answers because I've tried to teach you Trevor when it's multiple choice for Pete's eggs glance at your answers for help Right, Maria. What's this question asking me to find? Then what? Find the word then then what it wants me to get the y by itself Okay, I'm that you know what I'll eventually do that now I look at this equation this expression that they gave me are there logs in every single thing No, no log there. You know what I would do first I'd get all the logs to the same side because that's how I taught you to solve equations then algebraic Which would probably try the same strategy So the very first thing that I would do is plus the log base three of x over I have no idea where we're going I do notice in my answers. What don't you see in any answer anywhere a log? I Have a feeling will probably write something as an exponent at one point along the way. I do see some exponents So I'm gonna write this as the log base three of y plus the log base three of x equals c Carly I have no idea what to do now, but I'm stubborn. I'm clever. I'm a good academic student. I refuse to quit I've got to try something What might you try what log log can you suddenly now apply? I try combining them because I notice my bases are the same. I don't think that's a coincidence Adding two logs is the same as what okay instead of writing yx all right xy alphabetically How does that help? Or does it? I'm stubborn. I'm clever. I'm an academic student Rhett. There is no way. I'm allowing myself to freeze up by the way I'm trying to show you how I would do these on a test. I would be stubborn However, if I couldn't get it within 60 seconds, I would circle it and come back to it later I'm not gonna spend 20 minutes on one question. That's also bad test writing So I've gone through the rest of the test. I've come back to this one because it bugged me I got to try something. What else could I do if you know one, you know both So you ready Spencer you're going to once again tell us what that is as an exponent What to the power of what equals what Maria? What did you say this question was asking us to do? Oh? Which really means isolate don't get the wide by it. How can I get the wide by itself? Ah? I just got my nerdy adrenaline rush. I see where we're going okay That's one type of if then question The other classic one is it where they give you two log expressions Like example two here says number 17. Yeah, it was number 17 on the provincial exam about 15 years ago Says if log base three of x equals 15 then the log base three of one third x is equal to Haley can you recognize this as an if then question see the if and see the then now sometimes there won't be a then There will just be a comma because when you put a comma in English It sort of implies that but almost always they'll do an if then okay? My strategy is always start with the then This thing try turning it into the if oh and there's apparently Four answers and the answers are all numerical with weird, but we'll see and I Have three things once I look at the then that I check for first thing I always check first is base change are my bases the same in the if and in the then Okay, then I'm not going to use base change If not, I would but Joel they would pick this question very carefully My second base would be a base nine or base 27 or something that works well with a three But base change no Then I look and I ask how many logs do I have in the then if I have one term I break it up if I have more than one log I combine them I think here. I just have one single solitary log So my first strategy is going to be I'm going to break this up. I Have a one third and an x mathematically. What's happening between them? So what will that become outside the log? This is going to be the same as the log base three of one-third plus The log base three of x. Oh Ryan can you read this to me, please? Base three base three big say it again properly log base Can you read this to me? What is the log base three of x the same as according to this statement? Oh kind of nice Hey, hmm What's the log base three of one-third work out to in your head without a calculator? Not one log base three of three would be one. It's a log base three of one-third Negative one elevator. Yes. Yes. Yes. Yes This whole mess works out to that and I'm gonna argue that even mitzvah can get that one in its head mitzvah Add those two together. What's the answer? Kind of see how we're approaching these by the way You need to practice these and there's a bunch on the great big log review that I gave you number three This is actually from an Alberta Provincial exam on the Alberta exams. They had numerical response This was a written if I gave this to you it would probably be a multiple choice question But still we'll handle it if log base B of a equals point eight two Then the value of log base B of B over a correct to the nearest hundredth is Isabel see the then I'm gonna work with This thing flat. What's my base here? What's my base here? Not using the base change law so I always check that first because it's easy to spot Do I have more than one term or just one term? One term break it up more than one term to mine So one term I'm dividing what's dividing inside a log the same as black So you're telling me that this is the same as the log base B of B minus the log base B of a I Agree. Oh Emily, what's the log base B of B with authority, please? One oh and Emily According to this question, what can the log base B of a all by itself with no exposure to anything be replaced with? Apparently, this is one minus point eight two And I think it's fair for me to expect you to be able to subtract two decimals from one in your head with no calculator I would expect you to be able to do that without a calculator to point one eight Yeah, how many sig figs that's physics a couple more. I think three more two more number four What used to be number 32? If the log base two of B equals C then the log base four of B equals Hmm if the log base two of B equals C then the log base four of B equals and And then the options are C over to C squared to see or root C I'm guessing a two is gonna show up somewhere in the answer Down hit there Can you see what I'm gonna try what's screaming out to me as my first plan of attack here? Dominique You're right louder When I say louder, I actually mean louder not the same volume seen one act one and take two and action Dominique base change Can't you see how you can glance at it and tell hey two and four ah, but notice what they picked for the bases They didn't pick two and seven They picked two and four or they might have picked two and eight or two and sixteen or two and thirty two But I'm gonna tell you they're not gonna pick two and 20 They're gonna contrive these So I'm gonna rewrite that second the then as base To if I rewrite this as base two this is going to be the log base two of B Over the log base two of four now It's a base change law and you absolutely want to make sure you know this one now Brett can you read them and can you read me the very very top right here? I Disagree, you know what that really is and oh What is the log base two of four Emily? Hey? Didn't see that coming and tell about one line ago, but hey Starting to get the hang of these a little like there's a there's a science and an art to these and really all I can Do is show you a few and say here's a bunch knock yourselves out last one if log base five of X equals four point two six What's the value of log base five of 25 X squared? Try this one on your own base change No, oh one big log break it up Ryan here was my strategy Multiplying same as adding oh I notice the X doesn't have a squared here. I don't want the X to have a squared And log base five of 25 that's two Plus two log base X of sorry five. Oh log base five X. Oh, oh That's a four point two six with a two in front of it. Oh and a two plus in front of that. What's your homework? I gave you this a Logarithms review quite some time ago. I think did I not the great big unit review? We're approaching the end of the unit For more lessons, I think so I'm gonna start to give you a few questions from here Specifically I'm gonna give you some if then questions that you can start whittling away at now These are not all for homework right now These are due the day of the test, but if you're wanting some more practice on these types of questions Now if you can't find this great big one jot down the question numbers elsewhere and hunt for it later Or I have a few extras still buried away somewhere number five is An if then question Number 15 is a disguised if then question Number 20 is a disguised if then question D5 is a disguised if then question 38 is definitely an if then question 51 52 58 70 sort of It's a simplify log question, but that's okay. I thought there was one more. I knew there was at least one more 85 103 Then you come to some written questions If you're wanting more practice solving the logarithmic and exponential equations Lesson last or two days ago and three days ago, which I kind of yelled at you last day saying was really important You can try something like number seven number nine 11 12 those are logarithmic. Here's an exponential equation number 15 16 is another log equation 18 is another log equation Gives you some to try is all that do first of all try the workbook identity stuff It's good practice then you can start whittling away at this and you have a take home quiz Okay