 All right, mark your quiz. Number one, so we started this unit out, lesson one was a review of exponents, and I'm not actually going to ask you to simplify exponents on your test, but what I said was you want to be comfy with exponents to do the rest of this stuff, like question one. This was lesson two. Lesson two, we solved this type of equation. What type of an equation is this? This is an exponential equation. You know how I know, because the variable is sitting in an exponent. And what was our strategy? We said we wanted to write this as, do I have one base equals one base, are my bases the same, then I can equate the exponents. That's our goal. So I'm going to write this as 5 to the 2x minus 3, I'm good with that, but I'm going to write this as 5 to the negative 3. Trevor, do I have one base equals one base, are my bases the same, then I can equate the exponents. I can solve 2x minus 3 equals negative 3 plus 3 to both sides. I think I'll get 2x equals 0, oh, I think I get x equals 0. So if you got that, 2 out of 2, otherwise I would probably give you a half mark for that line, a half mark for that line, a half mark for that line, and a half mark for the answer. B, B, here is what I cannot do, here is what I cannot do, oh, 16 times 8, that's 128. You know why I can't do that? Because there's exponents on these, how dare I multiply before I've dealt with the exponents. Saw some students in my last class, they wanted to make that 128, no, no. Instead, what I'm going to do is I'm going to write this 16 and this 8 and this 2 all as the same base, I think base 2. I'm going to write this as 2 to the 4 to the x times 2 cubed to the 2x plus 2 equals 2 to 3x minus 1. Itzel, do I have one base equals one base, say no, in fact I have two bases on one side. Oh, but these two bases are the same, oh you know what else, first of all I'm going to do power to a power. I'm going to write this as 2 to the 4x, 2 to the 6x plus 6 equals 2 to the 3x minus 1. Itzel, do I have one base equals one base? Are my bases the same over here though? What am I doing between the bracket mathematically, adding, subtracting, multiplying or dividing? Oh, when I'm multiplying powers and my bases are the same, what do I do with exponents? I add them. As a matter of fact, this is going to be that plus that, gathering like terms. This is going to end up being 2 to the 10x plus 6 equals 2 to the 3x minus 1. Katie, do I have one base equals one base? Are my bases the same? Then I can equate the exponents. I'm going to solve this equation. I'm going to minus 3x from both sides. That's going to become a 7x, and at the same time I'm going to minus the 6 over to this side, and that's going to become a negative 7. Oh, I think x ends up being negative 1. Two marks. Otherwise I would give you a half mark for that line, probably a half mark for that line, probably a half mark for the equation and a half mark for the answer. Number three. Okay, kind of cool. I'll write this as 3. I have power to a power, so it's going to be x squared plus 2x equals 3 to the negative 1. Roxanne, do I have one base equals one base? Are my bases the same? Then I can equate the exponents. I'll get x squared plus 2x equals negative 1. Oh, what kind of an equation is this? It's a quadratic? How do I know? It's got a squared. How do I solve the quadratic? Oh, the first thing we always did was make it equal to 0. And then we either factor or quadratic formula this. This one actually, Amanda, I think factors. I think it factors into numbers that multiply to 1 and add to 2 or positive 1 and positive 1. Amanda, what are my roots? I think negative 1, yes? And I think negative 1 twice. In fact, I think last year, if you remember in math 11, you called it a double root. There you go. Kind of neat getting a quadratic out of an exponential. My little math nerd heart like that. Two marks. Number four says evaluate. These ones, I would expect you to be able to do in your head. I would expect you to know that the log base 2 of 32 is 5. You might need to count on your finger for the powers of 2, but I expect you to know your powers of 2. And I would hope that you would say, well, if it was log base 3 of 81, that's 4. Log base 3 of 1 over 81, that's negative 4. Number six says, using the definition of logarithms, what's the value of x in each? Where is the x sitting in number 6? Oh, you know what I'm going to do here, Andrew? If I know 1, I know both. I'm going to recognize that if they tell me a log, I know the exponent. If they tell me an exponent, I know the log. I'm going to write this as what to the power of what equals what? x cubed equals 8. Oh, what number cubed equals 8? What's the answer? 2. Now, what if you didn't know the answer? How do I get rid of a cubed? How do I get rid of a squared? Square root, how do I get rid of a cubed? So if you didn't know the answer, cube root both sides. But here, I'm just going to jump straight down to saying, hey, x equals 2. I guess one mark for that, and one mark for the answer. And now what this is really saying, it's all is 1 third to what power equals 9. That's what it's saying. If this was a 3 instead of a 1 third, 3 to what power equals 9? What about 1 third? Elevator, negative 2. One mark, one mark. But if you got the right answer, you get 2 out of 2. Could you give yourself a score of there, please, out of 12? Shannon, question? You good? Give yourself a score out of 12. Make sure your name is on it and pass them inwards. As usual, I'll start out by saying any questions from the previous day's homework. Aside from the two that I already told you, I'm going to be going over, for sure. Which ones would you like me to go over? Yes? 5B? OK. Thank you. 5B are my bases all the same. OK? Means I can combine these. But first, I need to get rid of the coefficients and move them up as exponents. In other words, this is going to need to move to there. This is going to need to move to there. This is going to need to move to there. I'm going to write this as the log base A of P to the 1 third plus the log base A of Q to the third minus the log base A of P to the fourth. Now I can combine them. And the short version that I gave you was positives on top, negatives on the bottom. This is going to work out to the log base A. There's going to be a P to the 1 third on top, a Q to the third on top, and a P to the fourth on the bottom. I'm not quite done. How many letter P's do I have on the top? 1 third of one. How many P's do I have on the bottom? Four of them. How many left and where? You know what? Here I'm going to need to find the common denominator to do this one. I'm going to have to rewrite this as the log base A of P to the 1 third, Q to the fourth, all over P to the... How many thirds is four? Are you ready, Nicole? This is by the way where your generation sucks at fractions. Don't worry. By the end of the year, you'll be mediocre at fractions. I promise. What's the denominator on this four? It's invisible. Over 1. So what's the denominator here? What would I multiply a 1 by to change it into a 3? What would I multiply a 4 by then? How many P's on top? 1 third? How many P's on the bottom? 12 thirds? How many thirds left and where? Thirds on the bottom. Should be what it says in the back. I think a hope, I think a hope. Is that what it says in the back? There you go. Next. Yes, Roxanne. Tricham. Yes? Not done yet, but is that legal? What is the log base three of 27? With authority, please. Three to what powers? What is the log base three of 27? You're right. With authority though, not as a guess. Three. In other words, instead of writing this as 1 half times the log base three of 27, I think what you're saying is it's 1 half times three. Roxanne, three over what? Don't make me write it, because I don't want to have to wash my hands, but three over what? Top times top, bottom times bottom. Yep. Usually I'll move the exponents to the front because then what's inside the log is easier. Roxanne, what is 1 half when it's an exponent? Which root is that the same as? If this had been an 81, I would have done square root of 81 first, nine, log base three of 92. I would have done it that way. It's a matter of convenience. Usually moving the fractional exponent to the front makes the math easier, because what's inside has no root, but if it would have worked out evenly, I'll do the root first. Is that OK? Next. Really, really, really, really, really? By the way, so far, how many questions have we done, two or three? You've asked almost everyone that my last class did too. That's why I'm pausing, because I know which ones you're going to have trouble with, I think. 10 C. Yep. Are my bases the same? OK. Well, I don't like the way this is written, because here I got a root. Here I got a fraction. First thing I would do, Nicole, is I would move that up to there so I can combine them. So I would write it like this. But I would also write it like this, Nicole, because the 1 half power is a root. I either want all roots or all fractions. And since the first one was a root, I'll write it as a root and see if I spot something like that. I just don't like to mix and match. Why make it tougher? What's subtracting two logs the same as? So if I hear you correctly, this is the same as the log base 2 of root 6 over root. What's 6 divided by 3? What's root 6 divided by root 3? Root 2. This is log base 2 of the square root of 2. Is what? Yeah, because it's 2 to the 1 half. It really inside there. 2 to what power equals the square root of 2? 2 to the 1 half equals the square root of 2. That's actually a bit of a review of math 10, where you did a big unit on radicals. By the way, when you're dividing square roots, it's numbers by numbers, roots by roots. If you have this, look up, look up. If you have 4 root 10 all over 2 root 5, it's numbers by numbers, roots by roots. Next, nothing before 11 and 13? OK, 10e? Yes, because that's what my last class asked, too, is why I paused. Are my bases the same? No. OK. I might have to pull up the base change law. I don't know. Let's see here. I do notice that I can write this as the log base c of a to the 5th over a. Are you OK on that, Sabrina? Subtracting 2 logs is the same as dividing. How many a's on top? 5. How many a's on the bottom? In fact, what does this simplify to? Log base c of what? How many a's on top? How many a's on the bottom? How many a's left? I can't hear you. I'm sorry. Sorry. How many a's on top? How many a's on the bottom? So how can I have 5 a's left if I have 5 on top and 1 on the bottom? 4, isn't it 4? Is 5 a's on top, 1 on the bottom? 1 would cancel. Yes? And I have 4 left? Yes? Moving further that way. Yes? Yeah, yeah, yeah, yeah. So this part here is, are you OK with that now? Because you didn't seem to, yeah? What's the log base a of a? No, too slow. What's log base a of a? What's log base 2 of 2? What's log base 5 of 5? What's log base a million of a million? What's log base car of car? What's log base a of a? Is that what it says in the back? Yeah, OK. By the way, if they really want to be to write this as a single log Sabrina, is 1 the same as the log base a of a? Is 1 the same as the log base c of c? If I wanted to write that as base c, because it's a 1, I could also have written this as the log base c of c. And now, what's between the two logs adding? These would be multiplied. It actually, if I wanted to write it as 1 log, would be log base c of a to the 4th c. Is that OK? Sorry if I came across as too intense. A little bit much caffeine this morning. Any others? Then I write this one here. I would consider G fair game as a nasty, though. I would consider this an A minus level question, but not beyond what I'd feel comfortable. I throw this as a multiple choice question on a non-calc section, probably. I would not make this a written worth four or five months. Blow my nose. That's going to be our haircut, too. Well, this is a power with an exponent. This is a power with an exponent. This is a power with an exponent. Except, holy, what's the base in all three of these? Route 2. If my bases are the same and I'm multiplying, what do I do with exponents? What about when I'm dividing? I think this whole thing is actually route 2 to the power of log base 6 of 27 plus log base 6 of 16 minus the log base 6 of 12. Does that make sense? Haven't done anything new? I've got to try something. That would be a little stubborn. Yeah? Oh, see that little exponent there in Nathaniel? Are all the bases the same? I think that means I can combine them. Adding, oh, positives on top, negatives on the bottom. I can write this actually as route 2 to the power of the log base 6 of 27 times 16 divided by 12. Now, I'm going to go to my calculator in a second. But Stephanie, before I do, Roxanne, before I do, what's my base on the log? I'll bet you a million dollars. This is either a 6 or a 36 or a 216. I'll bet you it's not a 10, because they probably wanted to make this work out evenly. What is 27 times 16 divided by 12? Ah, I told you. So if I hear you, you're saying this is route 2 to the power of the log base 2 of 36. Oh, base of 6 of 36, Mr. Dewick. Nicole, what is the log base 6 of 36? Are you saying this whole thing is just route 2 squared? Trevor, what's route 2 squared? Careful. What's the square root of 2 squared? 2. Did you figure that out or did you snag that from Tyson over there? OK. And number 13. This is your first exposure to what we call if then questions. Because this question says, if blah, blah, blah, blah, blah, then you're going to know and love and hate these. There's an art and there's a science to these. We'll be practicing. In fact, we'll probably spend one whole day practicing these in a couple of lessons from now. But for now, find the then and ask yourself and tell me, what are they asking me to find? Steph, what are they asking me to find here? Then what? Read the question. What are they asking me to find? You're louder, please, with authority. You're asking me to find the log of why. OK. What did they give me? What's the if? Ignore the greater thans. What's the equation that they gave me? The expression that they gave me? Steph, ignore the greater thans, I said. What's the expression that they gave me? What's the equation that they gave me? A bit further. The whole equation, please. This is the only equation in here. What's the equation that they gave me? Yes, because I said ignore the greater thans. You started to read those. I don't want those. That's just condition. If this, what's that? That's what they're saying here. Write this down. Accept, stop. Leave a little space after the equal sign. Steph, this is an equation. You know how I know this is an equation? Two things. There's an equal sign, and there's something on either side of the equal sign. So when I said to you, what equation did they gave me? Next time, you'll go hunting for what? An equal sign, right? In math 8, you learn that you can add both sides, five to both sides of an equation. You can multiply both sides of an equation by two. In math 11, last year, you learned that you could square or square root both sides. You know what else you can do? Log both sides, but it better be both sides. You've got to do the same thing to both sides. How did I know to do that? Look at your answers, the four answers that they gave you. What's inside every, what appears in every single answer somewhere? Yep. OK. Steph, what did they ask me to find here? OK. Oh. What's happening between the x cubed and the y mathematically right there, adding, subtracting, multiplying, or dividing? Amanda, what is multiplying two logs inside the log the same as doing outside the log? Or in other words, how can I break this up? It's the reverse of what we did last day. No, it's times right now. You guys stay here. You guys stay. Hang on. You guys stay here. Stay here. Patience. Something times something is the same as what? So if I hear you correctly now that I cut off your attempted interruption, if I hear you correctly, I can rewrite this as, yes? Amanda, what were they asking me to find? No, it's not what they're asking me to find. What were they asking me to find? I got a pussy. They're not asking me to get what y is equal to. Ah, what are they asking me to find? What log y is equal to? How can I get the log y by itself here? Yeah. Minus this over. Log z minus the log of x cubed equals log y. Now, I glanced at my four answers. I don't quite see that. But look carefully at your answers. I sort of see it. They've just gone one step further. So look at your answers here. And which of those is the correct answer to the expression log y equals? What'd they do for b? Oh, they took this exponent and what? To the front. What were you going to interrupt me about, or did I clarify it OK? Adding outside is multiplying inside. Multiplying inside is adding outside. What did we do here? We took the log of both sides. And that's a wonderful segue into the next lesson. Can you turn please to the next lesson, which I think is just page 135, if I recall. And first, I have to pause for a second. OK, take a home quiz away and look up please. All right. We've been looking at questions like this so far. How would I solve this question here? How would I solve this question here? We just had it on our quiz, boys and girls. How would I solve this question here? No, I would not. We just had this on our quiz. How would I solve this, folks? What would I do with the 16? Not two. Gosh, people, be with me here, please. You would do this. Yes? Yes? Yes? Remember this from five minutes ago? Yes? You better, because it's on your test. Here's the question we're going to ask today. Now what? Can I write 15 as 4 to some power? Not in my head. If it was 4 to the 1, I could do that. If it was 4 squared, I could do that. In fact, I'm pretty sure the answer is it. That looks like a 6, Mr. Dewick. I'm pretty sure the answer is somewhere between 1 and 2. How can I solve an equation like this, where there is no common base? And now, Andrew, you maybe might be able to deliver the punchline that you suggested about 30 seconds ago that I said say what? Do you remember that? Not adding the logs, taking the log of both sides. What we're going to be looking at today is solving exponential equations without using a common base. And to do that, we're going to take the log of each side. Let's look at example 1a. It says, express x as an exact value, and then to 3 decimal places. Solve, it says, 4 to the x equals 12. Where is the x sitting? Or what kind of an equation is this? Stephanie, why? Don't get this mixed up with something like this. We did those last, you take the fourth root of both sides. You're not going to be using logs or anything. It's not that it has an exponent. It's that the x is sitting inside an exponent. How will I solve this? I'm going to take the log of both sides. Screen froze? Great timing. Take the log of both sides. Ryan, why is this so nice? What can I do with that x now? Andrew, I can take it from upstairs where I cannot do anything with it, and I can move it to the front, and it's now on ground level. I can rewrite this as x log 4 log 4. Mr. Dewitt, not base 4. x log 4 equals log 12. Folks, can you just stop talking please? Why is that so nice? Because now I can get the x by itself. What's happening between the x and the log 4 mathematically, adding, subtracting, multiplying, or dividing? Times x, so how do I move the log 4 over? Hang on. x equals log of 12 divided by log of 4. Or, Jessica, they may write it this way, the log base 4 of 12, because that is really what you would type that to get that. These are the exact value answers. If they ever say solve using and give me an exact value, which is what the first part says, that's what they mean by an exact value. It means get as far as you can without a calculator. But they also set in the second part to three decimal places. So, OK, what is the log base 4 of 12? Or, what is log 12 divided by log 4? What is the answer to three decimal places, please? To three decimal places, please? Can't be 1.79, it's got to be 1.79 something. 1.7, what are the three decimal places? Come on, people. Log 12 divided by log 4, 1.792, yes? There, did you say that? I just heard a 2, but I didn't know where it came. I needed the 1.792. I never did hear that, sorry. Anyways, that's the answer to three decimal places. How do I solve an exponential equation? Take the log of both sides. How do I know it's an exponential equation? The x is an exponent. You're really going to get something much more like question b on your test, not question a, a, too easy. b, two different bases, each with different x's. See it? What kind of an equation is this, Carson? How do I know it's an exponential? Because where is the x sitting? Exponent, OK? By the end of the year, you're going to have half a thousand equations. So you ought to be able to figure out what type it is to know your strategy. This is an exponential because the x is an exponent. How am I going to solve this? No, I'm not going to move. How am I going to solve this? One step first thing, log both sides. Now, I've taught math 12. You guys are probably around my 30th class. You may have noticed most of the time as a math nerd, I try to encourage you to take shortcuts to try and do some stuff in your head. There is so much room on these to make dumb mistakes. There is so much room to make dumb mistakes that I'm going to tell you do not, do not, do not take any shortcuts. These will take about up to 10 lines to solve. It's worth doing each line step by step, trust me. Let me show you the first mistake kids make. So this is excellent. Mr. Dewick now says I can move the exponents to the front, which is true. I can move that to there, and I can move that to there, and I see this all the time. Anybody see where I've made my mistake? This is wrong. What about the minus one? I don't think that's it, really, it's part, sorry. This is written incredibly sloppy and wrong, and I'm going to make a dumb mistake on my next line and give you a hint it's the right-hand side where the mistake occurred. What was the exponent on the right-hand side to begin with? How dare I write that without brackets? That was the exponent, was it not? I see that all the time though. I understand, in fact, I have to be honest. I'm a little flabbergasted that nobody picked that up. My previous classes, everybody's like, no brackets, do it. Oh, someone said that back there? Should I say it louder? Sorry, I didn't hear. Absolutely, why? Well, for starters, it's sloppy notation, but it tells me what to do now. Look up, don't write this down. If you had this, 2x minus 1 times 5, what would you do now if you were trying to solve that as part of an equation? Get rid of brackets and distribute. OK, I have a 5 there. It's a log, big whoop. If I don't have the brackets there, the number of kids that just drop the 2x down and don't realize there's going to be a log attached to it, which there is going to be, and then I have to stop marking and give them 0 for the rest of it because everything else is garbage. Drop the 3x log 5 down, and I get 2x log 3 minus 1 log 3, except you can't make me write the 1 because it's invisible. Harvey, how many x terms are there in this equation? 2. How many would I prefer? Well, if you had x's on both sides of the equation in Math 9, what did you try and do? No, you're telling me that when you had this, you started dividing now? You got all the x's to the same side, yes? How would I get the x's to the same side? I think, whoa, that's no good. That's like the fourth time in five years that that program has crashed. It never crashes. All sorts of wonky things happening. How would I get? We're back, yay. How would I move this over then, Holly? Subtract, or if it was negative, I wouldn't subtract. I would add, but I'm going to subtract. I'm going to get all my x's to the same side. That's going to give me 3x log 5 minus 2x log 3 equals negative log 3. In fact, I would argue that so far, after this line, it's all been Math 9. Ugly Math 9, yes, but Math 9. Holly, how many x terms do I have here? How many would I prefer? It would be wonderful if there was some kind of a grade 9 mathematical operation that I could pull out of my back pocket that would somehow allow me to change that from 2x's into a single solitary x. Do you remember me saying that phrase before somewhere this year? What was, what did we do here? Math F word, not the F word you're thinking of. No, Andrew, no. Do I have one fraction equals one fraction? So how dare I even start to suggest cross multiply? How can I turn that into a single solitary x instead of a pair of yucky x's? First and loudly, factor, factor what? Factor an x out. Look, look, look, look, look, look, look. There is an x in both terms, yes? If I factor an x out, I'll have, I'll have 3 log 5 minus 2 log 3 equals negative log 3. And Kirsten, how many x's do I have now? Just one. Oh, what's happening between the x and the bracket mathematically, Kirsten? So how would I move that ugly bracket over? What's happening between the x and the bracket? Multiplying. So how would I move this ugly bracket over? Divide. It's ugly, but it's still math 9, right? If you, oh, multiplying, divide. x is going to be the negative log 3 is going to be on the top. And on the bottom, I'm going to have a 3 log 5 minus a 2 log 3. Or, now that's the exact value answer. Staff, they may write it like this. Amanda, where'd the cube come from? Yeah, and where'd the square come from? Although, you know what? I doubt they'd write 5 cubed. They'd write 125. And I doubt they'd write 3 squared. You know what they put there instead? I'm not. In other words, if you solve, and it's multiple choice, so you're looking for an exact value answer, and you can't quite find your answer, start applying the log rules. It's probably there somewhere in disguise. Now, that's as an exact value, Ryan. What about to three decimal places? Folks, here's Mr. Dewick's advice. I've marked this question twice on provincial exams, and I've marked about 1,000 papers. And it was amazing, Katie, how many kids could get to this line and couldn't get the right answer because they hadn't practiced on their calculator. So can every single one of you, every single one of you, every single one of you get your calculator out, please? In my heart of hearts, I should say no. Where's yours, Tyson? Try typing that in. I'm not going to show you how to type it in. I want to see if you make the most obvious mistake or if you fix the mistake. Try typing that in. I'll freeze the screen. I'll do it here, and then I'll show you what the answer is. You should get that. Negative 0.4175503838...... How many of you got that? Now, the rest of you, how many terms are there on the top? One, how many terms are there on the bottom? Then you better put the whole denominator in brackets. So if you didn't get the right answer, type with me. Here's how we would type this. Negative log three, and then we got to close off the logarithm bracket, divided by, and we're going to open up a bracket for the entire denominator. Three log five, oh, it just opened up a log bracket so I better close off the log bracket. Minus two log three, oh, it just opened up a log bracket. I better close off the log bracket, and now I'll close off that denominator bracket. There's your answer. There's your answer. In fact, I think that's so important. I'm even going to do this. I'm going to totally try that again because it's supposed to work differently. I want to click right here. There is what I wanted to clip in. There, out of the way. But I'll put that in the lesson. What you typed in to get the correct answer. Brackets, brackets, brackets, brackets, brackets. Oh, what if there was two terms on the top? Brackets around the top. Negative four point, sorry, negative point four one, I guess eight if I round off properly, yes? This question here, which took about seven lines, I would consider a C plus level question. One base, different base, different exponents. I sort of like this question. I really like this question. I like this question. I like this question. This is a nice question, C. What's the difference between B and C? C has two different bases. Does B have two different bases? Yes. C has two different X exponents. Does B have two different X exponents? Yes. C has a coefficient, that too. And that's going to make this an awful lot tougher. No way around it. Here's what you can't do. Here's what if you do this, I have to give you a zero out of five marks. And the test is out of about 50. So one of these will be worth about 10% of the test, just one question. Amanda, here's what you can't do. Oh, this is six to the X minus two. Is that X minus two on the two? Then how dare you multiply before you've done exponents? Tyson, what type of an equation is this exponential? How do I know without having to think about it? The variable is an exponent. You know what my first step is going to be? Take the log of both sides. I have to move this down, oh well. Log of two bracket three to the X minus two equals the log of seven to the X. By the way, when I don't write a base, what base are we using here? Yeah, we're using the common base. You could do L and the both sides, and still get all the same answer, but I don't want to terrify you. All right, put your pencils down. Here is the most common mistake, number two. And here is where I have to give somebody a zero. If they do this, I can't mark any further. They go, okay. Can anybody spot what's wrong with that, because that's wrong. For a candy, I'm curious. I didn't have one student suddenly see it last class. Why is this incorrect for me? Times what to the three first? The way I've written this, again, I've somehow said that that X minus two is on the two. Not on the two. Not on the two. Because can you see, Cassandra, I'm gonna get a six popping up again, and if it was wrong to get a six here, just as wrong to get a six here. By writing this, I'm telling people, oh, that X minus two was on the two, and it's not. So, what the heck can I do then? And this is where somebody saw it last class. What the heck can I do here? What's that? What's right next to it? No, it's not right, what's right next to it? What mathematical operation is happening between the two and the brackets? What's multiplying inside the log the same as outside the log? I gotta break this up. This is going to be the log of two plus the log of three to the X minus two equals the log of seven to the X. I have to break it up. But now look, can you clearly see the way I've written this, that the X minus two is only on the three and not on the two? Now I can move exponents to the front brackets if they're binomial exponents. Now I can do that and that. I drop the log two down plus X minus two log three equals X log seven. The rest of this, you should be able to figure out. Let's see. I have brackets. Go look at B when we have brackets. What did we do? What did we do? Look at B, find the line where we have brackets. What did we do? I heard it. Distribute, okay? So we're gonna go chunk, chunk. We're gonna get the log two just dropping down like a domino, but I'll get X log three minus two log three equals X log seven. Okay, we got rid of brackets. What did we do next? Again, look at what you wrote for B and see if you can figure it out. I don't think we factored yet. Polly, get the X's to one side. I heard somebody say collect like terms. Technically that's not what we're doing. We're gonna get the X's to the same side by plusing and minusing. I think the easiest way, by the way, I could minus this here and move those. The easiest way is just move this X over to this side. I think at least the amount of work. So I'm gonna get log two minus two log three equals X log seven minus X log three. Now what did we do? I heard someone say it, but factored. Because how many X's do I have? Two, how many would I prefer? One, I got them on the same side. I'm good with that. Let's factor out the X. What's happening between the X and the bracket? Multiplying, so Steph, how can I move the ugly bracket over? In fact, I'm gonna tell you, I think, I'm pretty sure, I think that X equals log two minus two log three all over log seven minus log three. Now that's the exact value. Oh, although Cassandra, if this was multiple choice, they might have written two log three as a log of nine because they might have moved the two up onto the three and made it three. I'll look for those things if I can't spot my answer. So that's an exact value. Try getting this on your calculator. Oh, how many terms on top? Two brackets. How many terms on the bottom? Brackets, you get this? Negative 1.775145987 blah, blah, blah. That's actually only part one of the lesson. Well, no, I always do this over two days. The next part of the lesson is we're gonna solve equations with logs in them, but for now, we're solving equations where the X is an exponent. How can you cheat? Well, it's not cheating. How can you check your answer? So Katie, you just finished my test and there's 15 minutes left. Are you gonna hand my log test in? Say no. You know what you're gonna do instead? Show you. You're gonna get out your graphing calculator. I'll show you how we can solve B on a graphing calculator to see if we made any mistakes. So get your graphing calculator out. Go look at question B. Don't press Y equals and clear any equations you have here. This says five to the three X equals three to the two X minus one. I'm gonna graph the left side. I'm gonna graph the right side. I'm gonna graph the left side as Y one. Five to the power of bracket, three X, I gotta use brackets for my exponents so it knows all of that is in the exponent. And three to the power of bracket, two X minus one close bracket. And then just in case you've messed around with your windows, why don't you all press zoom standard? The zoom button and then choose standard. And you should get something that looks like this. Now it's not very pretty, but you know what I'm looking for here? Where are these two graphs? Do what? Cross. Second function, trace, calculate. Intersection. First curve, enter. Second curve, enter. Guess, just use the guess that they gave you. See if you can find that negative 0.4175504 and then we know that we did it right. Can you see if you can get that on your own? Second function, calculate, which is second function, trace. Intersection. First curve, enter. Second curve, enter. Third curve, or guess, enter. So you have to get some batteries. Are we right in B? Let's try doing the same thing for C. So I'll clear whatever equations I have here. Two times bracket, three. Close bracket to the power of bracket. X minus two, close bracket. Seven to the power of X, close bracket. Graph left side, graph right side. Hit graph. Oh, I got a closed bracket there that I don't need since there's only a single solitary X. I don't need any brackets because it knows that that's a bracket. That's up top. If you hit graph, you get this. Yes? Just to show you how quick you can be. Because we're only graphing two things, as it turns out for the intersection method, everything that I want is the default value. In fact, look up. Here's how I do this, watch. Second function calculate, intersection, enter, enter, enter. That's how I do it, because it's first curve, second curve, guess, and all of those are what I want by default anyways. And is the answer negative 1.775, 1.4? I am right. I am right. What's your homework? Two things, take home quiz, and question number one, all. Question number two, all. And question number six. Your homework is one, two, six. One, two, and six. Sorry? I already said that. I said your homework is first of all, take home quiz. And then question one, question two, and question six.