 So homework from last day, two C. Now there's two ways to do this. The first is in your head, and this is how I would do it. And in fact, after today you'll learn a much better way to do this question because today we're looking at log law's shortcuts. First of all, LM is base what? This is really saying the log base E of E squared is what E to what power is E to the two power. E to what power is E to the two power? Okay, that's how I would do it. But if you wanted to do it on your calculator, you would press your LM button, well you turn your calculator on first of all. You press your LM button. I showed you last day where the E was, turns out the E is right above the divided by sign, second function divided by squared, and that gives you the value of two, but I wouldn't do it that way. I would just say it's the log base E of E to the two is E to two. Any others before I go to number eight? Yeah. I have no idea what you said louder. You can use either of those bases. What you can't do is use them both in the same question. In other words, if I said evaluate log base seven of 82, you could go log 82 divided by log seven. You could go ln 82 divided by ln seven. Same answer. You can't go log 82 divided by 82 bracket divided by ln seven. That won't work. In fact, you can use any base, base 10, base 10, base E, base E. If you had a different base on your calculator, some calculators might have a base two because that's also a common one. You've got to use that. Any others before I do number eight and number nine and number 10? Number eight, who asked me number eight? Let's do this as a calculator question, but I would actually probably do this in my head because you can, but for now, how would I type that? Log eight divided by log of root two, close off the square root, close off the log. With my base change log, or I could have gone ln ln plus two times, now log base nine of three would be log three over log nine, or do you guys know what is the log base nine of three? Nine to what power equals three to the one? It's square root, right? Which is what as an exponent. For what it's worth, I would go, that's a point five, point five, which it is. I notice a nine and a three, so let's see a nine and a three alarm bells are going off in my mind. They're related. I can probably do something with it. In fact, I'll be honest. I probably could have done something with the square root of two and the eight. I know that root two to the sixth power is eight, two to the third, square root of two to the sixth. Just good at square roots. Anyways, the answer is seven, which is what it says in the back. Number nine and number 10 are interesting. Now, there are better ways to do, in particular, number 10. But for now, I haven't given you all the tools and shortcuts. I noticed number nine, Eric, and number 10 are very similar. They're both one log equals one log, except in number nine, the X is inside the log. And number 10, the X is on the base of the log. See the very subtle difference there, which means I'm going to end up using two different strategies. In the first one, pretty much whenever the X is inside the log, if I want to get the X by itself, Ryan, I write it as an exponent. I know if I have the log base A of B equals C, that automatically means that A to the C equals B. This is the same thing, but uglier. I'm going to write it as seven to the power of what? Seven to the power of what? Log base four of 60 equals what? X. Now, why did I do that? Ta-da! The X is by itself. The left side is ugly. I won't dispute that. But wait a minute. Would I do this on my calculator, and then just go seven to that answer, and in fact, that's what I'm going to do. I'm going to do the log base four of 60, log 60 over log four, and then seven to that power. The answer is 313.29, oh, it says to the nearest whole number, doesn't it? X is 313. I sort of like this question in that I guarantee, Joel, I'm going to give you a question where the X is inside a logarithm, and you better clue in that, how do I get it out of a log? We said, if you know one equation, you know both equations. If you know a log, you know the exponent, if you know the exponent, you know the log. That's almost always one of my first strategies. Put your pencils down once you've written this down. I moved to number 10 in Vlad. I tried the same thing at first. I went, okay, this is going to be X to the log base 12 of three equals the log of 27. I went, bleh. If that was X squared, square root both sides, and I'm done. If that was X cubed, cube root both sides, and I'm done. It's X to the bleh, I'm not going to help you. I said, okay, now what? The only other thing I've learned is last day's lesson, base change. I said, I'm going to try writing this whole thing as base 10. This is actually the log of 27 over the log of X. This is actually the log of three over the log of 12. Since I wrote no base, that's all base 10. Now what? I had no idea at this point, but I'd written something. I felt better. I got to try something. I'm going to be stubborn and clever. I said, well, I have one fraction equals one fraction, why don't I cross multiply? Get rid of the fraction. So I did. Then I was going to write times log three, and I stopped and I said, hey, wait a minute. Why don't I just divide by log three and get the log X by itself, because I'll bet you I'll want to do that later on anyhow. Physics 12 students, I just moved up to agony. Why is that helpful? Where is the X sitting now, inside the log, just like question number nine? What did I do in question number nine? I'm going to try the same strategy. What's my base here? 10. It's going to be ugly, but doable. I think I'm going to have this, 10 to the power of log 27 log 12 over log three. That equals X. I think to evaluate this Mitsu, I would crunch the entire exponent first, log 27 times log 12 divided by log three, and then I would go 10 to the power of, and I would hit my answer button, which is second function negative on the graphing calculators. Holy smokes. It works out evenly. Kind of nice. X is apparently 1,728. Oh, and Vlad, I could check this, by the way, log, I could plug the X into the original equation, log of 27 divided by log of 1,000, what was it, 704, excuse me, gives me that, and then I could go log of three divided by log of 12, also gives me, oh, I'm right, got the same answer, both sides, pretty quick check, easy check, yay. Next lesson, turn the page, lesson six, this is the whole point of the unit. If you get today, the rest of this unit will make sense, for the most part. If you're zoning out today, you in trouble. So turn your brains up, ramp your brains on, turn the crank in your mind, get the little hamster running on the wheel, whatever you feel that you need to do. Says evaluate the following, and I don't think I need a calculator for this. What is the log base two of 16? You better be able to do this. What's the log base two of 16? It's four, plus, what's the log base two of three, ah, two of eight, three, there's a Freudian slip, you say one thing but you mean a mother, four plus three is what, okay? What's the answer? Seven. Now, I notice here we have the same numbers, but instead of adding outside the log, we're timesing them inside the log. If I go 16 times eight, I think, I think, I think, I get 128, 16 times 10 is 160 minus 32, 128. What is the log base two of 128? Fingers, two, four, eight, 16, 32, 64, ooh. You know what the log base two of 128 is? Also, seven. What do you notice? Same answer. Warmup two, evaluate the following. Now this would be four, take away three, the answer is one. And again, I notice that they're doing the same thing except they're subtracting outside the log. Here they're dividing inside the log. What is 16 divided by eight? That I can do in my head. What's 16 divided by eight to two? This is actually the log base two of two. What is the log base two of two? Same answer as well. This brings us to our power law and quotient law, the product, sorry, power, the product law and the quotient law. And these are important. How important? You need to know these. I'll even be more specific. If you don't know these, not only are you not passing the log test, the odds are pretty good. You'll flunk the course. Generally, I find the kids that can't master this sometime around Christmas, I'm signing a piece of paper. Do you need to memorize these back and forth? No, I would say if in a week and a half when you glance at this, this looks like Greek, then you're in trouble. But I think you'll memorize this naturally by practice. Here's what it says. If you're adding two logs with the same base, that's the same as multiplying them together inside the log. Or if you're subtracting two logarithms as long as the base is the same, it's the same as dividing them inside the log. Now why is that so handy? Put your pencils down and look up. This suddenly allows you to evaluate in your head a whole bunch of previously unsolvable or undoable logarithms. For example, the log of four plus the log of 25. Now I didn't write a base. That is my base when I don't write a base. Can I do the log base ten of four in my head? No. Can I do the log base ten of 25 in my head? Am I adding two logs? That's the same as this. What is four times 25? What's the log base ten of 100? The answer to this is two. You can't get it from here, but it sure pops out of here easily. Four, the log base four of 48 minus the log base four of three. Can I evaluate that in my head? Can I evaluate that in my head? Are my bases the same that I can combine them as one single log and you know what, subtracting is the same as inside log? Turns out this is the same as the log base four of 48 over three. What is 48 divided by three? Don't think it's 12, otherwise I've really botched my math here. This is the log base four of 16, which I can evaluate. What? Suddenly you can combine logs and turn them into hopefully easier expressions. Example one says evaluate the following using the product law or the quotient law. Are my bases the same? Subtracting is the same as dividing. This can be written as the log base two of 12 over three. Eric with no calculator, putting your calculator down, what is 12 divided by three? What's the log base two of four? In fact, I'm going to modify this rule even easier. I'm going to say if you're combining logarithms, positives go on the top, negatives go on the bottom. Top, top, you know where the two is going to go? On the bottom. In fact, a negative in front of a log acts sort of like an elevator as it were and causes this number to move down below. Yes, there's a reason why I brought that in three weeks ago. I can combine this as a single log rule by writing this as the log base six of, there's going to be a nine on top, an eight on the top, a two on the bottom. What is nine times eight divided by two, simplified two? Nine times eight is 72 divided by two is, by the way, what's the base that they gave me? What's the base that they gave me? If someone said 35, I'd be saying you're not a smart test writer because if it's base six, I'm pretty sure the answer is either six or 36 or 260 and otherwise I can't do this. It's 36. What is the log base six of 36? I get the answer is always two, the two examples that you did and the answer is just always two. I just realized the two examples that I did with you 30 seconds ago, both worked out to two and these both worked out to two. I don't want you to spot something that's not there. Something said that Emily, because I've only asked you to memorize fairly small powers, I bet you the answer is rarely going to be 57 because you don't know what two to the 57 power is or three to the 57 power. The answer will often be lower than 10 because I've got to keep my numbers reasonably small. See, what do you think I'd do with this thing and the brackets before I went any further? Front door bomber if you had Mr. Raqqa, distribute if you want to call it what it's supposed to be called. I would write this as the log base five of 10 plus the log base five of 75 minus the log base five of two minus the log base five of three. The basis is the same then I can combine them and here's what I've said to you, positives go on the, negatives go on the, so if I want to write this as a single log there's going to be a 10 and a 75 on the top and I guess between them I'll have to put a times sign. There's going to be a two and a three on the bottom and between them I'll have to put a times sign as well because they're both on the bottom so if they're both on the bottom they'll be multiplied together on the bottom. No calculators. This is fair game. Oh, by the way what's my base five? I'm expecting this to work out to either five or 25 or 125 or 625 but not much more than that. Just a guess. I have a 10 and I have a two what can I simplify that into? What goes, oh five and that would cancel. Oh and does three go into 75? Yes? How many times? In fact, I would argue this whole thing simplifies to the log base five of five times 25. What's that? What is, told you'd be one of those numbers, what is the log base five of 125? Not two. You're going to make it? Take your jacket off. I know but the jacket is keeping you warm and making you drowsy. When you're up late the last thing you want to do is wrap yourself in a blanket. Let me assure you, take the jacket off, assume that I have more experience in overheated classrooms than you do in your young lives. You know what? We need some extra. Can you tell me if you think it necessary? D. Okay. It's all one base so I can write this as a single solitary log. There's going to be a two on top, a three on top, a six on the bottom and an eight on the bottom. Because minusing means divided by and plus means times. By the way, when you're multiplying exponents and the bases are the same, what do you do with the exponents? You add them. Remember we said logs are inverses of exponents? When you add logs, what do you do inside the logs? You multiply them. It's exactly the same combination of operations. Oh, how many sixes are there on top? Can you see it? One. How many sixes on the bottom? In fact, Eric, I would do this. Come on. There. Sorry. That's a six and a six. What's left on the top? It's invisible but what's sitting there now? This is the log. You got to write the base. Base two of one over eight. What's the log base two of just plain old eight? What's the log base two of one over eight, negative three elevator? Very useful. What kind of a multiple choice question might you get during the page? I guarantee you're going to get something like this. I give you something algebraic and I say, hey, when you simplify it, what's it work out to? Now, I want to teach you how I would approach this multiple choice question. First thing I would do is I would glance at the answers. How many terms are there in all of my answers? Look at all four answers. How many terms are there? Two terms. Yes? A number and the log. How many terms did they give me in the question? Right? See it? One, two, three, four. That suggests to me I better combine them otherwise how can I get it down to two terms? The questions answers point a strategy out. Oh, are my bases all the same, Trevor? So I'm not going to be pulling out the base change law from last day. If they were different that would be a hint that maybe I was pulling out the base change law. You're going to reach the point in a couple of weeks where you have about ten tools at your disposal. If it's multiple choice often glancing at the answers Carly will tell you which tool to go with. So here I'm going to try writing this as a single solitary log. I'm going to write this as the log base two of, there's going to be an X, there's going to be a 2X on top, there's going to be an X squared on the bottom, there's going to be a Y on the bottom. Am I done? Is that answer appearing anywhere? Then apparently they want me to go further. What do you see? What else can I do? Eric, what? I think you're right. How many X's are on the top grand total? How many X's on the bottom grand total? How many X's are in any of my answers? How many X's left and where? That must be the right step. The answer is told me. So now I have this, log base two of two over Y. Eric can I do anything else with this fraction? How many terms do I have in my answer right now? One, how many terms are in the answers that they gave me? Two, oh, you know how I turn four terms into one by combining them? I think I can turn one term into two by breaking it up. What's dividing inside a log the same as outside the log? Do some of my answers have a minus sign between them? I'm on the right track. This is the log base two of two, take away the log base two of Y. I just heard about eight people go, aha, because they just saw the answer. I don't see the answer quite yet. Oh, wait a minute, Brett, what is the log base two of two? Do I have an answer somewhere that says one, take away log base two of Y? See how I let the answers kind of guide me, stubborn and clever a little bit, really also being a clever cast writer. Example three, evaluate that. Okay, are my bases all the same? Yeah, then I can combine them as a single log. No calculators. Can you tell me anything that goes into the top and bottom here? There's one that's kind of crying out to me, I think, an obvious one, and I would expect you to spot that. What number can go into the top and bottom here? Really? Five, yes? Isn't there a 15 and 105? Yes, five, yes, yes, yes? Let's do that. What's 15 divided by five? What's 105 divided by five? Well, what's 100 divided by five? 20, what's 105 divided by five? 21. Oh, now I see all sorts of stuff. What else can go into the top and bottom now? Seven, that's going to make the 14 a two, and that's going to make the 21 a three. Oh, how many threes do I have on top? One, how many threes do I now have left on the bottom? One, how many threes will be left and where? None. In fact, all that's left is what on the top? This whole thing simplifies to log base two of two. And what is the log base two of two, go pray, tell, Carly, one, is that neat? I like that one. I love questions that look ugly, and suddenly end up being like a one or a two as an answer, like aw, great. They got everything to cancel. B, I'm not going to do B because right now, the way that I would do B uses a rule that I'm going to show you in 60 seconds. So we're going to come back to B. Warmup number three says, show that two log base three of X equals log base three of X squared. What they're really saying is this, and this is the new rule, and this is why I'm going to use it in the previous question in a second. An exponent inside the log is the same as a coefficient in front of the log. That's what it's saying, but it wants us to prove it. Easy. Draw a little line down the middle of your page because for a proof, I want to make sure that I do one side or the other side, so I emphasize that by drawing a line. What's another way to write X squared? If I break it up, X squared really means what times what? That's the first thing I'm going to do. I'm going to write this as the log base three of X times X. I'll use the dot for times because otherwise it's going to look like X, X was confusing. What's multiplying two logs the same as doing outside the log? This is the same as the log base three of X plus the log base three of X, right? What number's in front of the log right here? It's invisible. What number's in front of the log right here? It's invisible. So listen closely, listen closely. What is one log base three of X plus one log base three of X? Two log base threes of X and wasn't that what I was trying to show? What I've really done is shown you for one example that as it turns out an exponent inside the log can be moved to the front. This is the power law. We have not proved it. We'll prove it formally at a later date. However, hugely important. It says that an exponent inside a log can be moved to the front which brings me back to this question up here. Here is how I would do this question. I would take this exponent and I would move it to the front. Eight log base four of two plus and I would take this exponent and I would move it to the front. Two log base four of one over eight. Four to what power is two? What is the log base four of two? Four to what power is two? Brett, why one half? Because the relationship between four and two is square root which is what as an exponent. The relationship between four and two is say square root. Square root which is what as an exponent? Say one half, thank you. So, if I hear you correctly my friend, that works out to that. What is eight times a half? What's half of eight? This whole thing works out to a four. Now, this one's a bit yuckier. Here what I would do is I would say, is that true? Yes, is that true? Pardon me? No or yes, can I? Yes or no? One over eight, elevate. Same as eight to negative one. But why did I do that? You know what I can do with that exponent? Move to the front. Minus two log base four of eight. Now, can I evaluate the log base four of eight? No. But the number four and the number eight, they do have a base in common. What base do they have in common? Two. I'm gonna base change this to base two. This is four minus two log base two of eight over log base two of four. What is the log base two of eight? What's the top? What's two times three divided by two? I think that's how I do that. I kept my numbers all small and I used my two times table for the most part. I did have to know that square root is a one half power, but have I been saying that you want to have that memorized and that cube root is the one third power? You want those ones in particular. Those ones show up fairly often. The power law, hugely useful. Eric question or Brianna question? Sorry, you good? So we have reached the point now where we probably have three or four different ways to do each of these questions. I'm gonna try and pick the easiest. Not always the quickest. Sometimes the easiest will be the quickest, but sometimes the easiest will be smallest numbers, maybe not the quickest. So evaluate four A. The first thing I would say to myself is can I evaluate log base two of 16? Does that work out evenly? And can I evaluate log base two of eight? Does that work out evenly? Then I'm gonna do that. If I couldn't, I'd combine them and hope that the combination would work out evenly. Like when we did the four and the 25 to get the 100 and the log of 100 worked out evenly. But here, I know that this is just a half times, times what? Minus one third times what? What is half of four? What is a third of three? What is two take a one? Mr. Dewick, I've got it, I've figured it out. What is it Eric? It's not that everything, the answer is always two. The answer is always two or one, Mr. Dewick. Because I'm looking, they're all ones. No, it's just a fluke. Don't look for patterns that aren't there. Now, B, what's my base in B? 10. Can I evaluate the log base 10 of five evenly? 10 to what power equals five? Can I evaluate the log base 10 of two evenly? No, then I want to try combining these. Ah! I can't yet. Before I combine them, I have to get rid of these coefficients. How can I do that? Move them back up to there. Are you a bit more awake? Take your, sweat and take your hoodie off if you're still tired. By the way, if you're ever sleepy and you're wearing a ski jacket and you can't stay awake in class, please don't take this personally, but you're stupid. Take your ski jacket off, like figure it out people. You're wrapping yourself in a blanket trying to stay awake, really? That's your strategy? Come on. How many of you drive? If you're tired and you're driving home late at night, what do you do? Oh, you open the window to get it cold so your heart rate increased? Really? Same principle. Spencer, what is five squared? And what is two squared? And I'm adding two logs. If my bases are the same and I add the logs, what's that the same as doing inside the log? Are you saying this is really the same as the log of 25 times four? Sandbilly, what is 25 times four? What is the log of 100? Mr. Duke, are you sure the answer is just not always one or two? No. Suggestions. How can I do C? You could either, and by the way, I'm going to argue the easiest way is gonna be bring the second three down. I'll show you why. This is what we're gonna write down. We're gonna do this. I always draw the little arrow reminding me I've done that. Or if I'm bringing it up, I draw an arrow in the other direction. Because if I do that, I'm gonna get the three log X is gonna drop down minus, oh, what is anything take away itself? What is three log X minus three log X? Can you say one? What's anything take away itself? Told you they weren't all ones and twos. Now, that's how I would do this. However, and don't write this down, another strategy would have been what Dominique suggested earlier, which would be move the three up there, you'd get this. Hopefully you would notice that that's something take away itself. But maybe you didn't. What's subtracting two logs the same as? Yeah, you could have gone. And what's anything divided by itself? No, what's anything divided by itself, kiddo? 10 to what power is one? You still get zero. You're gonna find now there's almost always at least two, often more than two different ways to get to the same answer. Almost always one will be cleaner. D, I would feel comfortable, totally comfortable, Kara, giving this to you on the non-calculator section. Do you know what 27 to the third power is in your head? Do you know what 81 to the fourth power is in your head? So now what? What tools do we have at our disposal? How many logs do they have in the question? How many logs do they have in the question? One, my rule of thumb is this, if I'm stuck and they gave me one log, I'll break it up. If I'm stuck and they gave me more than one log, I'll try and combine. That's my, don't know what to do, but gotta try something. They gave me one log, I'll break it up. What's dividing the same as? I would do this first. Oh, I still gotta do 27 to the third in my head and eight. Oh, or do I? What can I do with those exponents now? Oh, let's do that. So if I hear you, Alex, you're saying this is three log 27 minus four. Oh, I forgot the base three. Log base three of 81. Why is that nice? Ah, can I do the log base three of 27 in my head? What? This is three, and then a three. Minus, and then the four drops down in front. Can I do the log base three of 81 in my head? I can't, what? In fact, this is nine take away 16, which I would argue is now math eight. Spencer, what is nine take away 16? Power rule, very useful. You can imagine certainly in the pre-calculator days, would you rather do to the 15th power, multiplying something by itself 15 times, or timesing something by 15, the second one? Definitely the second one. It made math way easier. It turned exponents into times, oh, or even square roots. Instead of square roots, you would multiply by a half, which is dividing by two, which is way easier than trying to do a long calculated square root calculation. Turn the page. Example five, A and B are the same thing, and we're gonna memorize them, but we're actually gonna memorize right underneath where it says note with a little cop holding the sign, that's the rule. But first of all, take a look at example five. What can I do with that M? Why is that so nice? Because that gives me N log base six of six. And Brett, what is the log base six of six? In fact, this is really just N times one, which gives me M. Fair enough. What's the base of the exponent in this question? What's the base of the exponent in this question? What's the base of the log in this question? What's the answer when you simplify? No, what was the answer when we simplified? N. Are you ready? I'm gonna ask that again for a reason. Listen closely. What's the base of the exponent in this question? What's the base of the log in this question? And what did it simplify to? B. What's the base of the exponent in B? What's the base of the log in B? And what will it simplify to? Now, I haven't proved that, Emily. The proof is a bit convoluted. This is one of the few times what I'm gonna say to you memorize this. If you notice your base and your base are the same, the log base and the exponent base are the same, that thing drops to ground level and is the answer. As a rule, see this one I don't memorize because I can derive it almost instinctively, Brett. I would just move the N to the front and then I would say, oh, it's log base B of B, I won. We all got that one. This is the one that kids forget. This is the one I always throw on the test, by the way. It's the same rule in the different order. It's saying, look, whether you do the log first and then the exponent, or whether you do the exponent first and then the log, as long as your base and your base are the same, what's inside is the answer. It's this, so here's the second one. Two more, we're done. And by the way, this is it for log rules. I gave you the log definition. There's the product rule, quotient rule, the power rule. There've been a few other ones that fall out of them. For example, what's the log base A of A, one, that came out of the log definition? What's the log base A of one, zero, because anything to the zero power is one. Those aren't really rules. I usually can just re-derive them on the fly in a big hurry, it's pretty easy. But it's the big three here. Product, quotient, and power. And product and quotient are really the same rule, but inverses of each other, so that's it. Says write the following as a single logarithm. First thing I would check, are all my bases the same? What base is this all? 10, are there any other weird bases? If not, that would be a hint that I had to use the base change law first, and change it so all my bases were the same. And they would contrive it, they would pick it very carefully so that when you did that, the yucky bases would like cancel, or work out evenly, or vanish, or something like that. So are all my bases the same? Check, can I combine these? Yeah. Do I have any coefficients in front of the logs? Then I can't combine them yet. Oh, but what can I do with the coefficients? Move them up as exponents. First thing I would do is I would write, this is the same as log B, plus log D minus log E minus log A squared, minus log A squared plus log A for the one half. How's that? How about putting a five on the log E? What's adding the same as? Timesing, what's subtracting the same as? Divide, oh, you know what? I simplified that further. I said, actually, you just remember, positive's on top, and once again, a negative acts sort of like it ends up on the bottom. So I'm gonna write this as one big expression. This is the log of, and I'll draw a fraction bar, there's gonna be a B on top, there's gonna be a D on top, there's gonna be an E to the fifth on the bottom, there's gonna be an A squared on the bottom, there's gonna be an A to the one half on the top. Am I done? I don't think so. I have some stuff on the top and on the bottom. How many A's on the top grand total to the one half? How many A's on the bottom grand total? Two, how many left and where? As an improper fraction, how many left and where? I think three halves on the bottom, right? Two is four halves, we're doing some fraction math now, boys and girls. Two is four halves, we have a one half on top, we have four halves on the bottom. How many halves left and three halves on the bottom? This is gonna be, I would quit here. If it was multiple choice, they might have quit there, or they might have flour-powered the fraction exponent, square root of AQ, whatever. I don't like having exponents and roots in the same question, I like them all to be exponents or all to be roots. Perfect reference. Yeah, how do I know? No, if you leave it blank, what do we assume? Always. If you write log, it is base tan, unless you write anything else, it's base tan, by definition, by default. Because it's the common base, that's why we call it the common base, it's the most common one. Write the following expression as a single log. Ah, nothing Trev? Really? I got Ellen pretty good. You know what, I was aiming for him, but my aim must have just been a little to the left and down. Sorry. Hey, what would you do here? What do you think we'd do here first? Take an educated guess from your years as math nerds. I would get rid of the brackets. I'm gonna multiply the three over two onto there and onto there and the negative one half onto there and onto there. I'll do that first before I do any exponent stuff. This is three over two log base B of X plus, what is three over two times two? Three log base B of Y to the fourth minus a half log base B of X to the one half plus a half log base B of Y to the one third. Mr. Dewick, where'd that X to the one half come from? I told you, I like either exponents or I like roots, but never both in the same question. Minus, you know what? I did the same thing last class. Just as you raised your hand, something was bugging me subconsciously. Yes, you get a candy if you ask me later. Now what? Are my bases all the same? Okay, so I don't need to pull up the base change law. Oh, that's gotta go there, that's gotta go there, that's gotta go there and that's go there. I'm gonna get the log base B of X to the three halves plus the log base B of Y to the 12th because it'd be times the exponent power to a power minus the log base B of X to the, what's a half times a half? Oh, top times top, bottom times bottom, a quarter. Mr. Dewick, why didn't you move a negative one half up there? I hate negative exponents, why not just leave it down and just deal with the exponents? Minus the log base B of Y to the, what's a half times the third? Sixth, not too sick, top times top, bottom times bottom, right? Now we'll combine them, positives on top, negatives on the bottom. This whole thing is gonna work out to the log base B of, there's gonna be an X to the three halves on top. There's gonna be a Y to the 12th on top. There's gonna be an X to the one quarter on the bottom and there's gonna be a Y to the one sixth on the bottom. Am I done? Yeah. Do I have X's on the top and on the bottom? Then I'm not done. Do I have Y's on the top and on the bottom? Then I'm not done. How many X's do I have on the top, three halves? How many X's do I have on the bottom, quarter? How many left in the way? You know what, to do this, I'm actually finally actually gonna need common denominators, I think, to make this easier. Three over two is what over four. And 12, by the way, what's this 12 over? It's invisible. 12 over one is what over six? Well, you would multiply the one by six to get a six, same to the top. I think it's gonna be to the 72 over six. How many X's on top? Six fourths. How many X's on the bottom? One fourth. How many left in where, Brett? How many X's on top? Say it out loud. How many X's on the bottom? Say it out loud. How many left in where? Doesn't that make sense? Six fourths, one fourth, five fourths left. How many Y's on top? 72 sixths. How many Y's on the bottom? How many left in where? 71 sixths on the, yep. Oh, are both of these inside the log? I can't really tell unless I do that, right? Then you know they're both inside the log. There's your long rules. What's your homework? Well, you gotta take home quiz. And also, I assigned lots of practice you need to practice this, sorry. One, two, four, five, six, 10, 11, 13. One, two, four, five, six, 10, 11, 13. One, two, four, five, six, 10, 11, 13. One, two, four, five, six, 10, 11, 13.