 So, log's quiz number two says, use the log laws, now I said to you the curve ball is number four, so I'm curious, we'll see. Use the log laws to express each as a single logarithm and then evaluate, I can't do the log base four of thirty-two, it doesn't work out evenly. I can do the log base four of two, it's a half because two is the square root of four, which is four to the one-half and I could, you know what though, they're the same base, I can combine them. This is going to be the log base four of thirty-two times two. Hey, that's the log base four of sixty-four and Mr. Dewey graciously made me memorize certain exponents. I know that's a three, one mark. What base am I in question two if there's no base, what base are we? Yeah, we call it the common base, we got lazy, that's the one we're going to use generically. So this is going to be the log of four times five divided by two, which I'm pretty sure is the log of ten. What is the log base ten of ten? What is the log base five of five? What's the log base a million of a million? What's the log base gamma of gamma? It's got to be one whatever that gamma is unless gamma is a zero, which is not allowed as a base. Oh, this is the log base three of nine. We said cube root was the same as a one-third exponent and the man to the reason that's nice is I can now move that one-third to the front. This is one-third times the log base three of nine and I know that the log base three of nine is a two and I know that that's a fraction it's really two over what one I'm not going to write that and I know that to multiply fractions that's the easiest one it's top times top bottom times bottom. I'm pretty sure the answer is two-thirds. There's about three other ways you can get to that same answer that's how I would probably do it though. Question? No, oh you're looking kind of like maybe just waking up gotcha. Number five? Wait a minute Mr. Dewey, what about number four? Yeah I'm gonna do it last. Number five says evaluate okay this is the log base four of this is the log base four of there's a six on top there's a 64 on the top there's already a three on the bottom and there's gonna be an eight on the bottom as well and that's how I handle fractions by the way. By the way I'm gonna assume this is gonna work out to a four or a 16 or a 64 because otherwise why would they give me a log base four if I get an answer of 17 I've messed up somewhere. Let's see if this was on the non-calc section I would go six and three three goes into six twice and eight goes into 64 eight times really I have a two and an eight and ah good gosh this is the log base four of sixteen two. Andrew says I'm telling you every answer works out to two Mr. No just those will show up fairly often because there's only so many exponents and powers that have asked you to memorize. Example six the log base five of root 175 over root seven. Hey we did one kind of like this someone asked me about this in the homework and I said something like this what's 175 divided by seven I think 25 how did I guess 25 my base is five I'm not guessing 15 I'm not guessing 20 I'm not guessing 30 those I'm not guessing 22 those will be dumb guesses pretty sure it's 25 so if 175 divided by seven is 25 what's root 175 divided by root seven root 25 right you went to the next step but I'll write this this is the log base five of root 25 and Amanda what is 25 I could make this an exponent one half move it to the front but you know what the order that I do things in I do whatever is the most convenient sometimes I'll do the root first sometimes I'll make well here because like the cube root of nine didn't work out evenly that'll make it an exponent move it to the front this is the log base five of five which is one now before I do number four can you all do me a favor see where it says out of six at the top make it out of five because number four is gonna be a bonus question but I didn't want to tell you that I wanted you to deal with something that you thought was for marks and didn't know how to do because that may happen on the test what are you gonna do wanted to get used to that feeling of no I can do this let's see it says this log base three of four is equal to X express log base three of two in terms of X I think what I want to do is try and somehow write this as a log base three of four what's my base here Andrew what's my base here so I'm not pulling out the base change law I would if it was like a nine or a 27 or a one-third or I try base change and I'd hope to spot something what's inside my log what's inside my log not a two I see a four in disguise I think I can write this it so as is that not still technically the log base three of two now why is that nice what can I do with that exponent and I'll get one half the log base up not base four mister dude the log base three of four why is that so nice what is the log base three of four the same as when I see a log base three of four what can I replace it with when I see a log base three of four what can I replace it with when I see a log base three of four what can I replace it with I can replace it with an X what it says is a one-half in front and an X and you know what I've just done I have written this in terms of X you know what the log base three of two is it's a half X if the log base three of four is X then log base two of three base three of two is a half X and this is your final answer now first of all just out of curiosity who got that anybody didn't think so now these are fair game if they give you an if then question now the word then doesn't appear here but it's if blah blah blah then express start with the van and what you're gonna try and do is turn it into the if using log rules and then keep track of whatever extra stuff you needed to bring in to turn it into the if and move that to the front if you can or do something with it and you should be able to get an algebraic expression we're gonna spend a whole day on if then questions but that's your first one so if you be so kind as to give yourself a score not out of six but out of five that will be delightful and do wonderful and then pass them in can you turn to the back of your book where there is a blank page please can you turn to the back of the book where there is a blank page please and for the heading you can just write practice exponential equation practice exponential equations so here's what we looked at yesterday I'm gonna do this and then I'll take questions about the homework but I have a feeling by doing a couple of more examples I might answer some of the questions in the homework don't write this first one down don't write this first one down if I give you something like this don't write this down 5 to the 3x minus 2 equals 25 to the 6 minus x this one we can solve without logs how what would we do to solve this one without logs really we wrote a quiz on it quiz one okay we would write this as I'd say that's not a 25 that's a five squared I would tie it up then I would say do I have one base equals one base I would answer yes or my base is the same I would answer yes and then I would equate the exponents write this one down law 5 to the 3x minus 2 equals 15 to the 6 take away x what type of an equation is this it's an exponential you know why because the variable is sitting in an exponent how do I solve this we added one new rule to your list of equation solving rules we said there's something else that you can do to both sides of the equation you only do it when you have the variable as an exponent what do you do to both sides of the equation take the log of both sides and I yelled at you guys last class I said don't you dare take shortcuts way too much room to make dumb mistakes I'm going to be very meticulous mr. do it could I move the exponents down right away you know what you're asking for trouble and dumb mistakes because I always have a kid that does that and they almost always do it wrong and then I can't give the marks for taking the log properly because they tried doing two steps at once and since they got the step wrong I can't give them any marks for the line and they threw a half mark now I'm going to move the exponents down to the front ah but when I move these exponents down to the front what am I absolutely positively going to remember so I always make a little note like that with a little squiggly line to remind myself that they went down there you don't have to but it helps me and it's going to be 3x minus 2 log equals and in brackets 6 minus x log 15 why is that so nice what do I do now now who has solution manuals okay in the solution manual what I've noticed the author does is he actually evaluates the log of five right now as a number as a decimal and he carries it to about eight decimal places and he carries that to about eight decimal places he gets rid of the logs which is legitimate you can solve it that way but you can't get an exact value that way you can it'll take you right to the decimal value and so I'm sure the method I'm showing you gets you an exact value if it's multiple choice and you're having to just pick from an answer on a non-calculator section this is what the answer would look like so what am I going to do now I'm going to get rid of brackets and that's going to give me 3x log 5 minus 2 log 5 equals 6 log 15 minus x log 15 double check yes yes yes yes how many x terms do I have 2 are they on the same side well let's deal with that I think the easiest a man that would be plus this to that side and at the same time I'm going to plus this to log 5 over that I'm going to get all my x terms to one side and all my non x terms to the other side and I usually try and do it in such a way where I can get rid of as many negatives as possible because negatives are going to make dumb mistakes is that right by the way look up for second Sabrina you want to know the biggest most common sloppy mistake at this point people are so paranoid about the logs they let their guard down for the math 8 the number of times I see that they plus this one over they did that first and like yes I got it right and I'll just drop that there and they forget to actually plus or minus it over and then of course everything after that unfortunately as well how many x terms do I have holly I mean what I prefer one it'll be wonderful if there's some kind of a grade 9 mathematical operation that I could put on my back pocket somehow that would somehow help me turn this from two terms into one term what I can factor an x out and I'll get 3 log 5 plus log 15 equals 6 log 15 plus 2 log 5 I'm home free except for some reason I see a lot of kids freeze here because it looks ugly what's my final step how would I get the x by itself well first of all Shannon what do I want to get rid of on this side I think this big great big bracket yes what's happening between the x and the bracket so holly move it over yeah it's ugly fine but it's still math 8 I would argue x equals 6 log 15 plus 2 log 5 all over 3 log 5 plus log 15 now we're going to go to our calculator in just a second but I just want to show you different ways that this could be written could they move the 6 up there as an exponent and the 2 up there as an exponent and the 3 up there as an exponent yeah or even could they hey what's adding 2 logs the same as what's adding 2 logs the same as they could write this all as one single log of 15 to the 6 times 5 squared whatever the heck that is they could write this what's adding to log same as multiplying they could write this as one great big log of 5 to the 3rd times 50 whatever the the heck that is. I would never bother, I figure, I like this better. And I'm pretty sure on my tests, I never do do that. And I guess since there's no provincial exam this year, I don't have to worry about you guys running across that. Let's go to our calculators. How many terms on top? Brackets. How many terms on the bottom? Two brackets. I'm deliberately gonna freeze the screen. Try this on your calculator. I said, it's amazing how many kids can get to here and can't get the right answer. And then on a five mark question, you just threw away one mark out of five and it was cause you don't have to type. Not cause you can't do the math, that's silly. No, you know what? It was cause you were too lazy to practice during class. That's silly. 2.583099382, blah, blah, blah, blah, blah. Okay? Brackets around the top, Ryan. Brackets around the bottom. And every time you type a log, you have to close off a bracket for the log. There. And I'll even put what I did on my calculator. I would consider that a B level question. It's a question. What do I mean by a B level log question? Two different exponents and they're both binomials. Both two terms. What's an A level question you ask? Well, let's try one. Put an equal sign. Show you how I would make one up. Katie, give me a number between one and 10. Five. We just did five in the previous one actually. Give me a different one. Seven to the power of four minus text. I don't know. Just making up an exponent. Give me another number between one and 20 this time. 11 to the 3x plus one. Now that's the same as we just did. Two different bases, two different exponents, binomials. If I wanted to make this tougher in front of the seven, I would put a coefficient like six. Oh, and in front of the 11, I would put another coefficient like three. Actually, I just gained you. I just fibbed to you guys just a little bit, OK? See that three on the right-hand side? See it? What's it doing to the bracket? Multiplying, which means I can move it around. I can move it to the other side if I want to. How could I move it to the other side? Divide. If they gave me, if I was dumb enough to give you this, first thing that I would do is I would say, not do it. You're not going to get me. It's not two coefficients. Really, Mr. Dewick? Really, Mr. Dewick, the question you're having is, what is six divided by three? And I'm allowed to divide because there's no exponent on the six. I'm not breaking my bed math rules. Really, the question I would give you is this, 2 bracket 7 4 minus 2x equals 11 3x plus 1. If you ever have Cassandra two coefficients, you don't move one of them over to the other side. Just deal with one coefficient. But this, I would consider an A-level question. I am going to put one of these on your written section. Two different bases, two binomial exponents, and a coefficient. What kind of an equation is this, Carson? Exponential. How do you know without having to think too hard anymore? Variables and exponent. Now, if you can't remember the name of it, that's fine. But make sure you clue in, variables and exponent. My strategy is not going to be make it equal to zero. My strategy is not going to be trying factor. My strategy, first thing I'm going to try and do is do what? What's the first step? Really, if you're not sure, if you look at the previous example, what's the first thing we'll do on the first line? Log both sides. Do you remember what I said was the most devastating, the most common mistake here on the left-hand side? People move the exponent to the front now. And if you move the exponent to the front now, what you're really saying is that exponent isn't just on the seven, it's also on the two. We can't move it to the front now. What did we have to do? We had to take this log, which has two terms in it, and break it up. What's happening between the two and the bracket? By the way, I should probably technically write this like that so that you know all of those are inside the log, otherwise you might think it was log two times seven to the blah, blah, where the seven wasn't inside the log. I should probably do that just to avoid making a sloppy mistake later on. What did you say is happening between the two and the bracket? So if I break this up, what will it be? I can't tell whether you guys are quiet and don't like volunteering answers or are just completely lost. If you're completely lost, that's okay. If you don't like volunteering answers, get over it, please. What would I do here, Steph? I'll start calling names now. What did you say? It's multiplying, what's multiplying inside a log? The same as where, outside the log. So I'm gonna go like this. Log two plus log seven to the four minus two x equals log of 11 to the three x plus one. Now, Kirsten, I can move the exponents to the front. Now I can move the exponents to the, oh brackets, brackets, brackets, brackets, brackets. Now what? I would argue that now we've turned it into the previous question, a little uglier version, but I would go find the line with brackets on the previous question and I'd say to myself, self, what did we do? Oh, and again, I said to those of you that have the solution manuals, in the solution manuals, the author makes this a decimal, makes this a decimal and multiplies the decimal into the brackets, which is perfectly fine and legal if I ask you to solve it without an exact value. I would just carry about eight decimals to make sure I don't get a rounding off, Eric, as logs are yeah. But I would argue this is actually in some ways easier. I'll get this, the log two is gonna drop down like a domino. I'll get a four log seven. I'll get a minus two x log seven equals, I'll get a three x log 11 and I'll get a one log 11. Am I gonna write the one in front of the log? No. Holy now what? Look at the x's to the same side and get the other stuff to the other side. I think the easiest is gonna be move this guy to there by plusing it and move this guy to there, how? By minusing it. Again, it's amazing how many kids make sloppy mistakes on this line because they're so relieved that they've done the nasty stuff. Now it's like math, they move and they've let their guard down. Don't do that, I'll freak on you in love but I'll freak on you. In a nice loving future kind of a way. I'll get this, log two plus four log seven minus log 11 equals three x log 11 plus two log seven. Oh wait, Mr. Dewick, two x. That would have been sloppy, it would have been devastating, it would have been bad. Now what? Okay, it would be wonderful if there's something I've written on my memory and I can pull it in my back. By the way, I use these phrases to try and help jog your memory and give you a easier way to remember just like how to find an inverse with checks and y around, who's in physics. I don't know the normal force. Oh, but you know the punch line but I'm really trying to teach you to give you a little memory trick, right? I have, oh, alarm bell. Hey, that was a fun one last unit too, wasn't it? Yeah, it worked. Cosme trauma says Carson, well, it still worked. So I'm gonna factor out an x and I'll have three log 11 plus two log seven. Final step to get the x by itself is what? Divide. X is gonna be log two plus four log seven minus log 11 on top all divided by three log 11 plus two log seven on the bottom. Did I miss anything? I'm gonna double check to make sure I haven't made a typo. Nope, okay. Evaluate it. That's the exact value. Give me the answer to, oh heck, three decimal places. Unless by an amazing fluke this works out evenly but I'd be stunned if it did. I don't know, I'm just letting people try. Point five four eight, is that what you said? Anybody else point five four eight, point five four eight? Yeah. Steph got it? Oh, Steph, how could you cheat? You guys remember, I also taught you last day how you don't hand this test in early. You go back to these two questions and you graph left side, graph right side and find where they cross. Graph left side, graph right side and find where they cross. And if you get the same decimal, you smile and you say, excellent, I just know I'm getting 100% on one section of the test. That's a nice feeling. There's gonna be one, maybe two of these on your test. They're worth four to five marks a piece. So if there's two of these, that's 10 marks out of 50. That's 20% on your test. Yes, that's a nice feeling, okay. Any questions there at all? Now there may have been questions about the homework. I'm gonna deliberately say, I'm hoping I answered some of them right now. I will take questions when I see you in a week but we need to finish off the lesson from last day. So if you can be so kind as to go back to your workbooks and turn to page, turn to page, 126, page 136. By the way, if you're trying the graphing thing, try it later, but it works, trust me. Page 136. Oh, you might want to make, the page that we just did, that might be one, you want a dog ear, hey, that's part of your study notes, Mr. Duick has just told you two questions. You'll have those same two questions on your test with different numbers and different coefficients in front of the X's, but it'll look really similar to that, okay. Page 136. Carson, if we call an equation where the X is an exponent, an exponential equation, what do you think we call an equation where the X is inside a logarithm? A logarithmic equation. So part two, solving logarithmic equations, okay, right? There are many techniques for solving logarithmic equations. I want you to look at example two. There are two main types of logarithmic equations. Now what makes an equation a logarithmic equation? Where is the X sitting? Inside a law, okay. There are two main types, where there's logs in everything or where you have logs on one side and no logs on the other side. We're gonna start out with the same approach and then come to a fork in the road and use two different solutions at the end, okay. Can you see the difference though? Logs at everything, no logs on the right, right? The approach is the same for both of them at the beginning. I want to rewrite this as one thing equals one thing. Here I have two terms. I want to write this as one term equals one term. Oh, are my bases the same? What's subtracting two logs the same as? I can rewrite this as, have I written that as one thing equals, one term equals one term? Good, gonna do the same thing over here on the right. Oh, are my bases the same? I'm adding, what's adding two logs the same as? This can be written as, the log base five, I'll use a square bracket and then curly bracket x plus one, curly bracket x minus three equals one. Do I also have one term equals one term? But can you see the difference? On the left hand side, I have one log equals one log. On the right hand side, I have one log equals a number. Let's look at the left hand side. I have one log equals one log, the logs cancel. Well, okay, I should be careful. Technically what we're doing is we're taking the anti-log of both sides and recognizing that the quantity of the first log has to be the same as the quantity that the log canceled. If you have one log equals one log, same base, you can say my real equation is x over 15 equals 0.2. Now, I'm doing bad math right there because a math professor, when I say the logs cancel, would freak on me because it's like I'm dividing both sides by log when I say they cancel. No, what I'm really saying, Andrew, is look, if you have the log of something equals the log of something, the two somethings have to be the same. We're taking the anti-log on both sides if you really want to think about it. But I'm going to say the logs cancel because that's easier for you to remember. Oh, how would I solve this? What's the 15 doing to the x dividing? So how will I move it over? x is 0.2 times 15. x is three, three. Look at the right. Can I cancel the logs? Do I have a log on the right-hand side? Do I have a log on the right-hand side? Can I cancel the logs here? No. Now what? So here's the second strategy. This is the two types of equations. If you have one log equals one log, the logs cancel though, you're not taking the analog both sides, fine. I'm going to write this as an exponent. What to the power of what equals what? Use your log definition because if you know one, you know the other, write that as an exponent. What number to the power of what number equals what? I'll give you a hint. Your base is your base is your base is your base is your base. Five, oh, let's use blue, Mr. Dew, it can be consistent. Five to the power of, now I got to be honest, if you don't know the log definition, you're flunking the test. If you need to go back and really review your notes. The log definition said this, look up, if A to the B equals C, I automatically know that the log base B, sorry, base B, Mr. Dewick, you don't know the log, base A, nice try, of C equals B. That to that equals what's inside the log. Five to the power of one equals what's inside the log. Now, why is that so nice? What's happened to my logs now? Gone. In fact, what kind of an equation is this gonna be? It's gonna be a quadratic. You know how I know? Because when I multiply out the brackets, there's gonna be an X squared. I can do that much in my head. Let's do that. What's five to the one? Come on, people. What's five to the one? Five equals, and I get X squared minus two X minus three, I think if I foil that out, what kind of an equation is this? It's quadratic. How do I know? It's got a squared. How do I solve a quadratic? What's the first thing I had to do before I did anything else? Make it equal to zero, all minus five from both sides. Now what? Factor, or quadratic formula, but I'm gonna try and factor. Are there numbers that make it equal to zero? Are there numbers that multiply to negative eight and add to negative two? Are there numbers that multiply to negative eight and add to negative two? Yeah. So this factors into X minus four, X plus two equals zero. What are my roots? Positive four and? Now actually, for both of these questions, we're not quite done. There is one more thing you need to do. You need to check for what are called extraneous roots. You see, there are some numbers I can't take the log of. What can't I take the log of? Negatives or zeros? So double check. If I put this three right there, will that give me a negative inside the log? So I'm okay with that. I'm gonna put a little check mark saying that I checked it off. Let's check this one here. If I put a four inside here, what's four plus one? That works okay. What's four minus three? That works okay. If I put a negative two right there, what's negative two plus one? Can I take the log of a negative number? This one here, we reject. Much like when Ryan asks someone on a date, reject. It's always a basketball player. Ryan, sorry, always a basketball player every year. And this is how you're gonna remember it. You're gonna hear me say Ryan's date for the rest of the year and you'll know rejection. You can say much like when someone comes at you for a layup, you reject. You can use that one if you want, whatever works for you, but I gotta get something. This is an extraneous root. It pops out of our modified equation, but it does not work in the original equation. And I guarantee you on your test, I'm gonna give you some log equations with extraneous roots. It means you just gotta quickly go back and check. Plug what you've got into the log expression and if you get a negative inside the log or a zero, no, no. He's part of a proud tradition. It's been basketball players for about 10 years. Come on, bear with me. Example three, it says solve and verify. Well, I'm just gonna check for extraneous. I'm not gonna do a complete verify. Okay, log X squared minus log X cubed equals 10. First thing I wanna do is write this as one term equals one term. What's minusing the same as? So this is really the log of X squared over X cubed equals 10. Do I have log equals one log? Do the logs cancel? Nope, oh, I'll have to write this as an exponent then. What's my base here? What to the power of what equals what? This to that equals what's inside the log. 10 to the 10th equals, by the way, what is X squared over X cubed? How many X's on top? Two, how many X's on the bottom? Three, how many X's left and where? One on the bottom. Now it's amazing how often kids freeze up now that there's an X in the denominator. And I'm gonna tell you right now on your test there's gonna be one question where you get an X in the bottom. Math eight, yeah, it's math eight. How can I solve this, boys and girls? Andrew, this is cross multiply because I have one fraction equals one fraction. Do I really need to put the over one? Come on, I don't wanna wash my hands just by writing that out. See the over one there. Anytime there's a number, recognize there is a fraction over one. This is one fraction equals one fraction. I can cross multiply. I'll get 10 to the 10th times X equals one. How would I get the X by itself? How would I get the X by itself? Divide by 10 to the 10th. And I could write out the one with 10 zero, but 10 to the 10th is big. I'll just, X is one over 10 to the 10th, which is probably scientific notation. One divided by 10 to the 10th power. Yeah, it gives me an answer. One times 10 to the negative 10, which is really what it is. One times 10 to the negative 10. Actually, I don't like this question because the numbers are so big. I would have preferred a base of two there and a base of two there, and maybe a four there or something like that, something smaller, but whatever. Two types of logarithmic equations. Logs in everything. Write it as one log equals one log, logs cancel. Logs in some stuff, not logs in some stuff. Get your logs to one side, get your non-logs to the other side, and write it as one term equals one term. Write it as an exponent. Turn the page. By the way, I'm skipping the example four. What's your hint that example four might not work out evenly? Two decimal places, okay? I said our strategy is, first of all, is there logs in everything here? Does every single term have a log in it? Nope, I got that three. Let's get the logs on one side and the non-logs on the other side. The first thing I would do is plus the three over. This is gonna be two log base three of x minus log base three of x plus three equals three. Now I want to write this as one term equals one term. Oh, wait a minute. These bases are the same, but I can't combine these logs yet because of this coefficient. But what can I do with that two to get it out of the way temporarily? Move it up as an exponent, right? I'll get log base three of x squared minus log base three of x plus three equals three. I bet you I'm gonna get a quadratic here because I'm seeing a squared popping out of here in the long run. I don't know, maybe it'll cancel, but we'll see. Now let's write this as one log equals, or one thing, one term equals one term. Shannon, I almost gave the pulse of punch line. What's subtracting two logs the same as with authority like you know what you're dividing? I can write this as the log base three of, there's gonna be an x squared on top, there's gonna be an x minus three on the bottom, and a three over here. Do I have one log equals one log, no? X plus three, thank you, sorry. So how do I get rid of the logs here? I write this as an exponent. What number to the power of what number equals what? This to the power of that equals what's inside the log. Three to the third equals x squared over x plus three. By the way, what is three to the third? Don't say nine, what is three to the third? 27. Now what? Is this not one fraction equals one fraction? Cross multiply. I'm gonna write this as 27 times x plus three equals x squared times one. What kind of an equation is this quadratic? How do I know it's got a squared? What am I gonna do? I think get rid of brackets, make it equal to zero, and then I'm gonna try factoring, but since it's said to two decimal places, I'll bet you it's gonna be quadratic formula. I bet you. Maybe not, maybe they're just trying to throw me off. 27x plus 81 equals x squared, x squared minus 27x minus 81 equals zero. Are there numbers that multiply to 81 and add to negative 27? Multiply negative 81 and add to negative 27. I'm getting nothing. Nine and nine, no. Three and 27, no. 181, no, I think that's it. So how can I solve that? How can I find those roots? Quadratic equation. x equals negative b plus or minus the square root of b squared minus four ac l over two a. Remember that, bad boy? Now, pause. Everyone get your graphing calculators out because there is a chance that your graphing calculator has a built in quadratic solver. So get your graphing calculator out. And what you want to do right now is press the apps button, which is right below the x button. It's blue on the TI-83s. I think it's gray or brown or beige on the, press your apps button right now. And you wanna see if you have an app called Pauli Smolt P-O-L-Y-S-M-L-T. Or you might have Pauli Smolt two. Who has that? Can you all check? Because many of the school ones will have it too. Press the apps button, Sabrina. And do you have something called P-O-L-Y-S? What do you have? Yeah, is that your calculator or my calculator? Oh, okay, that's why. Those of you that don't have it, in the next few days when I say get to work, bring me your calculator and I'll install it. I'm okay with you using technology. Those of you that do have it, run it. Press the number next to it or whatever. Okay? And then what does it say on your screen? I think it says press any key. So press any key. Yes? And then I think you have several options. And the first one is polynomial root finder. Option number what? Let's press a one. And then, now if you're using Pauli Smolt version one, it says degree question mark. What's the degree of a quadratic? Two. And if you're using Pauli Smolt version two, I think it has the numbers from zero to nine and you can just use your arrow keys to pick the degree and there's a bunch of other stuff on the screen. Regardless, highlight the two. Yes? And then I think along the bottom, is there commands along the very bottom row of your screen now? I'm going from, by the way, I can't install it on here. So I'm having to go from memory. No? Sorry? Okay, hit two and hit enter equals. Do that. Does that take you to a new screen? Okay. That's the quadratic formula. See, look what they wrote up there. Except instead of using A, B, and C, I think they're using A2, A1, and A0. But it's still A, B, and C. Okay? So, what's A? One. Enter. What's? Look at the, they gave you a template there. You don't have to ask me. They gave you a quadratic template there. Yes? Yes? Top of your screen right there. They actually wrote it out for you to say, here's where the numbers go. So, what is it on your program? I don't remember. Is it A2 or A0? A2. What's A1? Negative 27. What's A? Negative 81. And then how would I solve? I think solve is the bottom right and it lines up with your graph button, doesn't it? That's the navigation keys that they're using. There's your roots. Who does not have the quadratic solver? Okay. You guys, for now, see this line right here? See this line right here? Graph left side. Graph right side. And you know what I'm looking for? Where those two graphs do what? Cross. I'll go zoom standard just to make sure I'm in a good window. Got a bit of a problem. It crosses there. I think it probably crosses way up here, too. But let's try it. Second function, calculate intersection. First curve, second curve. I'll move a guess close to here. Enter. Is one of the roots on the polysmoat, negative 2.724? Yes? Two decimal places. And I'm gonna go, so are y'all with me on the intersection method? Right? Those of you that don't have a polysmoat right now? Second function, calculate. Intersection. First curve enter, second curve enter. Now I want to see if there's a root over here off my screen, way up in the air. You know what? For my guess, I'm gonna type in positive nine because that's right there. And I'm hoping it'll sneak this way and find it. See how clever we can be? Let's see. Oh! It only found that one, is there only one root? There is a second root, what's the second root? Sorry? 29.7, so right now I haven't found it yet. I'm gonna have to go window and I'm gonna have to make my x max maybe 50 to see if this fits on my screen. Let's see. And I might need to make my y max 100 to see if I can get, well cross is there, crossing up there somewhere, I'm pretty sure. I might have to make my y max 300, I don't know. Crossing up there somewhere. I'm gonna see if it'll find it now. First curve enter, second curve enter. Guess, I'm gonna go, well I think I typed 50 as my x max, I'm gonna guess 49. I'm gonna have it start right here and see, oh there it found the 29.7. You can get there. But I'll be honest, I find the poly smelt easier. 29.72, why doesn't Ryan get a date? Look, look, look, look, look. If I put a negative right there, what's, oh that's gonna give me a negative log right there. Can't take the log, give a negative. One more and we're done. By the way, I took a long time on this lesson because I really wanted to make sure you grasped it. You're gonna have homework over the four day week but I don't see it for a whole week. You can spread it out and not have homework on your weekend. I just, last one. Okay, oh, is this a logarithmic equation? Yeah, you know how I know because the x is inside the log. So I'm gonna try and write this as one term equals one term. Are my bases the same? What's subtracting the same as? I can write this as the log base four of x plus one over two x minus three equals the log base four of eight. Do I have one term equals one term? Check. Oh, do I have one log equals one log? Then the logs cancel. No, I'm taking the analog of both, when they cancel. I can go, my equation is really going to be x plus one over two x minus, Mr. Deweyck, minus three equals eight. And again, it's amazing how many kids get here and they freeze. This is math eight. Is this not one fraction equals one fraction? It's an ugly fraction. It's not eight, it's over one. Cross multiply. You're gonna get x plus one times one which is just plain old x plus one. And you're gonna get eight times two x minus three. You're gonna get x plus one equals 16x minus 24. You're gonna get a 15x and you're gonna get a 25 over here. When you minus x from both sides and plus 24 to both sides, you're gonna get x equals 25 over 15, which is really in lowest terms, five over three. Oh, but we need to check for extraneous. I need to make sure that if I plug it into here and here, I don't get a negative answer. Five over three plus one, positive. Two times five over three minus three. Oh, positive. It works. What if it didn't work? What if I had no answers that worked? No solution. That's what I would write. No solve, no solution. So there's log equations. I already assigned number one and number two and number, oh, I think three, four. Yeah, G and H are nasty, but if you get those, you'll find the test easy, so I'll leave those. I skipped five, yeah I am. Ooh, those are yucky. I already assigned six, yes. Seven and eight.