 For the benefit of the people who aren't here who are listening to this on the internet, I've given out a practice test. I'll put this practice test as a PDF file in your block B or block D or block H online folder where it says click here for today's notes and hopefully the answer key will be this thing here. Number one, it says by the way we should be able to do numbers one through ten without a calculator so we're going to try to, where is the X sitting in number one? It's in the base. When the X is either in the base or inside the logarithm, what's our strategy? If we know one, we know both. That's our strategy. Now what does that remind you? Every log equation, excuse me, every log equation gen is actually an exponential and vice-versa. So I would try writing this as an exponent. My base is my base is my base. The answer is the exponent to the one half and that equals what's inside the log. Now how does that help? Some of you may already say oh a square root is one half power but just in case we didn't spot that, I now have a fractional exponent. What was the trick with fractional exponents? What did we do to both sides? Reciprocal power. Who's in my block H math class? So we haven't gone over the big quiz yet have we? So that's what you've been doing on Friday during class. I'll just put a video on and you'll watch the video, the TOC will put a video of the class of the quiz on. You'll find that also clears up a lot of stuff, okay? Nice thing is when I square a one half when I go power to a power, that's just a one and I get x equals 10 squared. Is that fair game to ask you to do without a calculator? Yeah. What is x? What is 10 squared? 100. Okay. Now this is a different order than what you'll see on the test. If you want a pretty good indication of the order, I'll ask you stuff on your test. I gave you a great big unit review that's an old test and I probably took great my new test, started with that old test to change some of the questions and change lots of the numbers, but probably the basic ebb and flow is probably fairly similar. Hit, hit, hit, okay. Number two, solve that. I don't see a 27 or a 1 over 9 or an 81. What do I see? I see threes. I would write this as three cubed to the 3n minus one times three to the negative two because that's what 1 over 9 is equals three to the fourth to the n plus two. I think now I would power to a power here and here. I'll get three to the nine and minus three times three to the negative two equals three to the four n plus eight. I don't like that left-hand side very much though, Leslie. It's got two terms, but what's happening right there, times. When your bases were the same, now I got to remember on math nine, what did we do with exponents when our bases were the same? Adam. In other words, I'm going to add nine n minus three plus negative two. I'm going to add nine n minus three plus negative two. When I gather like terms, I think I'm going to get this three to the nine n minus five equals three to the four n plus eight. Miguel, what's my base on the left-hand side? What's my base on the right-hand side? Do I have one base equals one base? My base is the same, then I can equate the exponents. You remember doing a little mantra. Again, I'm trying to help jog your memories. My equation that I'm actually going to solve is nine n minus five equals four n plus eight. I think I would minus four n from both sides. I'll get five n minus five equals eight. Now what? Plus five, I'll get five n equals 13. I think it's going to be 13 over five d. On your test, it's going to be some kind of on the non-calc section, solving an exponential equation where your bases are in disguise, but they are the same. I don't know if it's going to be quite this tricky. That was kind of the mini little, hey, what do I do with that? I think I kept it one term equals one term, not two terms equals one term. Number three, okay, here's an if, then question. Justine a little bit strange because they wrote the if afterwards, but I'm pretty sure we're saying break this up. Oh, no, hang on. I went too fast. Justine, what's my base in the if? What's my base in the then? Okay, because I do check for base change first because that's an obvious one to spot. Now if it's one log, I'll break it up. Hey, if it's more than one log, I'll try it because it's combining it. Here's one log. I'm going to break it up. Dividing is the same as, this is the log base m of n squared minus the log base m of m. I kind of like this, sorry, m, Mr. Dewitt, let's try that again, not nm. I do like this because Miguel, what is the log base m of m? What about this? Well, I would move the exponent to the front. Oh, but what is the log base m of n the same as according to this question? So if I hear you correctly, you're saying this is two times six minus one, okay, which is what? What if they did give you a base change question? What would that look like? What are you made of? First of the practice test, lose the chuckle and sit somewhere where you feel smart. So what if they did give me a base change log question? What would that look like? Well, I'll try and make one up. I've seen stuff like this before. If the log base a of six equals d, don't use it, equals c, and the log base a of d equal, or using numbers, Mr. Dewitt, is what happens when I try and make these up on the fly, of seven equals d, then the log base c of 42 over c squared, okay? So here's an example of kind of a weird one. What's your base? What's your base? Except I don't want to make this a c squared. I would like to make this an a squared, Mr. Dewitt. Try that. There. Make up on the fly, Jan, trust me. What's your base here, Jan? I rewrite this as base a, I think. Well, if I do that, honestly, though, I think what I would do first is I might break this thing up a little bit. This is the log base a of 42 minus, sorry, base a, Mr. Dewitt, base c of 42 minus the log base c of a squared. What I would say, you know what, 42, what's 42? That's six times seven. I think this is actually the log base c of six plus the log base c of seven minus the log base c of a squared. What base would I prefer to write this in? Base a. I would rewrite this as base a. This is going to be the log base a of a squared all over the log base a of c. This one is kind of nice because I can move the two to the front and I'll have the log base a of a with a two in front of it. What is the log base a of a? In fact, this whole thing just works out to a two over the log base a of c. What about here? This would be the log base a of six all over the log base a of c plus the log base a of seven all over the log base a of c. Do I know what the log base a of six is? Apparently that's a letter c. Do I know what the log base a of seven is? Apparently that's a letter d. Now, if I gave you this question because I just made this up on the fly, I would have to give you one more piece of information. I would say something like the log base a of c equals two and then you'd be able to replace that with a two and replace that with a two. You could do something algebraically with this and get an expression. I made this one up. This is not a great example, but there was some on your quizzes or in your notes or in your homework. In fact, the day we did the if then questions on that lesson, I had some that had good base changes and there are some in your big review. Am I going to give you one like that on the test? Well, it's fair game. Not like this because I made up a crappy question, a better question than that. But you can tell pretty easily when you're supposed to use the base change law because the bases are the same. Here the bases are different. Here the bases are the same, so no base change law. Number four, what the heck is that saying? Look at all your answers. Jesse, what letter do you notice is by itself and all of your answers? I mean by itself and all of your answers. Look at all your answers. What do they get by itself? What do they get by itself, folks? Hey, see, I didn't know what they meant when they said write an expression for A in terms of B and C, but when I glanced at my answers, I went, oh, they want me to get the A by itself. Is that a log right there? Yeah. How about right there? Yeah. How about right there? Yeah. How about right there? You know what? Let's get all the logs to the same side. Oh, and that's actually going to be kind of nice. I'll plus this log over to this side. So the first thing that I would do here is I would write this as 1 equals, even though we're trying to get the A by itself, that's going to come later. Let's get the logs by themselves first. This is going to be the log of B minus, and I would right away move the exponent back into the log, Kyle, because I have to do that before I can combine them, plus the log of A. Why is that so nice? Look at the right-hand side. Do they all have the same base? I can combine them. Positive's on top, negative's on the bottom. Adding means multiply. Subtracting means divide, but I even simplified it. Positive, put it on top, negative, put it on the bottom. I'll get this. The one's going to stay because there's no logs on that side, but I'm going to get the log of, you know, on the bottom there's going to be a C to the, let's try that again, Mr. Duke, a C to the fifth. On the top, instead of writing BA, Justine, do you mind? Can I write AB? That's kind of how, we're used to saying things alphabetically. Is that okay? And this is all inside one great big log. Now what? Where is the A sitting? Inside a log. Okay, we already mentioned this earlier. If my variable is inside a log, what's my strategy? If it's an exponent, I can move it to the front, but A is not sitting as an exponent. I'm going to write it as an exponent. Yes. If you don't want, you know both. What's my base here? When they don't write a base, what's my base? That's kind of sneaky of them because I'm willing to bet if they put a number there, you would have spotted it a little sooner. But with no base, sometimes we forget there is a number there, Jordan. It's a visible 10. 10 to the 1, oh, 10 to the 1 equals what's inside the log. By the way, what is 10 to the 1? 10. Okay, I'll look at that on my next line. And Jesse, what did you say? What are we trying to get by itself? A, we're multiplying the B. I think we'll divide to move it over. We're dividing the C to the fifth. I think we'll multiply to move it over. I think A is going to be 10 times by C to the fifth divided by B. Is that an answer that's sitting there somewhere? And especially because this is your first Math 12 test, and probably you haven't written that many tests that have a multiple choice section. I'm really trying to teach you how multiple choice questions, they are tougher, but there's huge advantages from the answers. You can often glean a lot of information. Like, oh, I'm trying to get the A by itself. Next page. Jen, you got a question? I don't mind re-explaining it. When there is no base, what's my base? If we just, in other words, there's a 10 right there. Shay Riley, could you report to the office now, please? Shay Riley to the office. Okay. Ask again, sorry. Did you see, I did my exponent thing, right? This to this equals what? Your base is your base is your base is your base. The answer is the exponent. And it equals what's inside the lock, right? Yep. And you're in a bit of a disadvantage to the rest of them. The blockage students, the rest of them got this beaten out of them when they wrote the big unit review quiz, because their stuff's at their fingertips a bit more. So yep, good question. Hey, five. Most of you probably aren't that quick yet. It is A. Really? Watch. I think it's A. You know what? This is a test. I might have made a sloppy mistake. I already told most of you that was my Achilles heel in high school. I'd get 99 out of 100 all the time. And it was always a dumb mistake. Too much of my head. Dividing by two, what's that the same as multiplying by? I'm going to write this as a half log A plus a half log B. Jesse, I would have glanced at my answers. And here's what I notice. How many log terms are there in each answer? Two, how many word logs do you see in each answer? One, so let me ask you again. How many log terms are there in each answer? How many did they give me here? I probably want to combine. Even if I really wasn't sure what the heck they were asking, I think I'd reason my way to that. So if I want to combine logs, can I have coefficients? No, move them up as exponents. By the way, Jen, what's my base here? Yeah, no number 10, right? A is not the base. Down there is where I put the base. Nothing, 10. Are my bases the same? Oh, what's adding two logs the same as? What is adding two logs the same as? Timing? Yep, I heard this. Sorry, I agree with you, multiplying. So I would do this. Now normally, this is as far as you can go. In other words, if we had a squared b cubed, we're stuck. But the fact that both of these exponents are identical, and not only that, I've tried very hard to pound into your brain that one half as an exponent is the same as what? Square root. This is actually the same as the log of square root of a, square root of b. By the way, do you see why I picked a as my answer? I mean, now we're multiplying inside the log. I'm not going to magically make it adding inside a log. It'll be adding outside a log. They're trying to trick me with b. I'm going to cross out that one, and I'm going to cross out that one, adding outside a log is with times inside a log. I'm not going to fall for that. That's sneaky of them. Real question is, what's a root times a root? Remember from math 10 when you're multiplying? What the rule is? It's numbers times numbers, roots times roots. What's the number in front of the square root right here? It's invisible. One. What's the number in front of this square root right here? It's invisible. One. So it's going to be one times one. I'm going to write that. No. But then it's roots times roots. Love. Root. Maybe. That's a little obscure. But that's it. Don't turn the page yet. Let's go back to number five. Supposing, supposing Justine, this was on the calculator section, and you had no idea what to do. But you've got extra time. You finished the test. You circled this one, and you just come back to this 20 minutes left. What are you going to do? Honestly, let A and B be actual numbers. I might let A equal seven, and B equal 20. I'm just making up two numbers. And on my calculator, I would go the log of seven divided by two, plus the log of 20 divided by two, and I would jot down that decimal, 1.073. And then I would type each of these into my calculator, letting A be seven and B be 20, and see which of these, as a decimal, gave me 1.073. Because if it works for numbers, then they're the same for algebra two. You guys haven't figured that trick out yet, by the way. For weird algebra ones, now the odds are pretty good, Jordan. I wouldn't give you this one on non-cal, but might. If it's a weird algebraic one, it's on the calc section. That's what I said on the calculator section. Make up numbers for your letters, but make up unusual ones. Don't use ones, because often they might cancel when you didn't mean them to. I would use twos and sevens and sixteens and thirty-nines, and just crunch the numbers on your calculator. If it works as a decimal, it's got to work algebraically. Some of you are just looking like, I never thought of that. You guys got to get better at test writing. Anyhow, I'm going to erase that, so I don't panic the kids that get this as an answer key. Take a look at number eight for a second. I know it's crossed out. I'm going to do this next unit, but I'm just going to show you domain for log graphs. What can't I take the log of? A negative or, I've got to be really fussy, or zero. If I hear you Troy, let me reverse the question. What can I take the log of? Are you saying that what's inside the log has to be greater than zero? That's what you said. That's my domain. Oh, I'm not done yet. Plus the ten over. Divide by five. You know the domain of that graph is? That. I'll be showing you guys that next unit, but for what it's worth, you guys don't realize how much I've snuck in kind of in the background that we can now apply in the next unit. That's on the page. Number nine says, which of the following equations may result in solving that? I guess this is their way of getting you to solve an equation without a calculator. They'll just say, get the equation and then quit. Okay. Oh, this is a log equation, isn't it? Our strategy here, Brett, was write it as one thing equals one thing. Are my bases the same on the left-hand side? Okay, so no base change law. And again, if I ever did give you a different base, what is my original base here? I'd make this like a base four, or a base eight, or a base 16, so that when you did rewrite it as base two, that four, or that eight, or a base 16 would become an actual number and you could do something with it. Don't think I did that on your test, though. I think I'd do that on one of your practice provincials. It's twigging with me, but it's not twigging with me like I've just seen it recently. So let's combine these. What's subtracting two logs the same as? Oh, but before I do that, I need to move this coefficient up to there, but don't I? This is going to be the log base two. What did you say subtracting was the same as? Dividing. Is that okay, Jordan? I would glance. I don't see it yet. Oh, Jordan, what don't you see in any of your answers anywhere? What's conspicuously absent that is in the original question? Logs. How would they get rid of the logs? If you know one, you know both, right? You see how, again, you can figure a lot up by glancing at your answers. Get that habit. What to the power of what equals what? Two to the third equals what's inside the log. What's the correct answer, folks? See it? Eight. I got a clue in that two to the third is eight. Oh, yeah. Do it. Did say there was certain exponents. We have to memorize. Number 10 says two students solved this using different methods. Their work is shown below. This is too much typing for me. I won't give you a question like this, but this is nice for us just to practice spotting the work and looking for common mistakes. Let's look at Jen's solution. Jen, let's look at your solution. It looks like you said, hey, that's base four. Four squared. That's base four. That's four cubed. Is this line correct? Yeah? Let's put a little check mark there to say, yeah, we're good with that. What would we do with these two here? Power to a power times them? What is two times three X? Sorry, what? Six X. What did they do to get the eight? They didn't go two times three. What did they do because they were in a rush? And went to the power of three, didn't they? So is this line correct? No. Oh, Jen. Sorry. Let's go look at Bob. I have a Robert in this class. I have a Bob in this class. But to me, the name Bob kind of conveys to me a certain kind of a, I am Bob. Justin, you can be Bob. You chose to take the log of both sides, which also works, although I wouldn't in this case because taking the log of both sides is a lot of work. So is this line correct? Taking the log of both sides. Is your X an exponent? Then it is correct. Practice test right there if you want to copy. Then what we, Bob, what would you do? That's good. Is this line correct? I disagree. I think they've made the classic mistake. You know what they forgot to do? They forgot to do brackets. Because look at their next line. They haven't multiplied. See, they should have done that. And there should be a log attached next to the X. Shouldn't there? Sorry, Bob. Oh, and it can't possibly be both. I should have crossed Jen both out as soon as I knew that you were wrong, right? Yeah, metaphorically, you know what I mean. Neither. I like that. I look at some common mistakes. You know, you can sit back there. I just grabbed the test. OK, or you can sit up there. Feel taller, don't you? It was funny. Miguel walked into class this morning. He sat there. I think, would you say, this is the happiest day of my life? Looking down on people. That's what most of us do. 13. Which of the following is equivalent to that? Jordan, what don't you see in any of the answers in number 13? OK. So apparently we can get rid of this. What might we try? Because I really don't know where this question is going. This is where we pull out the log rules like a machete and start whacking away. What log rule might we try first? Troy, try moving the exponent to the front. I don't know. Why not? If I do that, I get 4M log base 8 of 512. Does that work out evenly for the log base 8 of 512? Now it does. Mr. Dewick. No calculators. Put them away, Jordan. Mr. Dewick, you didn't make us memorize the 8 exponent. No, I didn't. Because you know some of them. Jordan, what's 8 to the 1? What's 8 squared? 64. What's 8 cubed? I don't know. But what did you say 8 squared was? Instead of going 64 times 8, which is what 8 cubed is, I'm going to go 60 times 8 because I can do that in my head. What's 6 times 8? What's 60 times 8? So I think 8 cubed is going to be close to 480. Is 512 close to 4? You know what? That's 8 cubed. The log base 8 of 512 is 3. So when I told you, Leslie, that you had to memorize a bunch of exponents, I didn't lie, but it might have been a fib. You can, if you're clever, really get away with only knowing some of the basics and then doing some rough guesstimation, draw some rounding off, and then say, well, it's got to be close to 512. Like, I know it's not 8 to the 4th because it wouldn't be 480 times 8. Nah! And I notice no decimals in my answer. This is on the non-calculator section. It's got to work out evenly. Otherwise, they wouldn't put it on the non-calculator section. Or are we on the calculator section now? I still would have done this in my head to summon her. How would you have done this with a calculator? Log 512 divided by log 8. Or if you're really a nerd, ln 512 divided by ln 8, and that would have given you 3 even. Yep. Okay. Write that as a single logarithm. Oh, I forgot to circle the answer. C. Sorry. Good point. Are my bases the same? Okay. Oh, but I've got coefficients. I'll have to move the coefficients into exponents and then combine them. So let's do the coefficients first. This is the log base A of 20 to the third. 20 to the third? Probably 8,000, but I'd have to check that. Plus the log base A of 10 minus the log base A of 16 to the 1 quarter. Are my bases all the same, Justine? Positive's on what? Negative's on what? Yeah. And this is where Greg has pointed out that we are in the calculator section. So you know what, Greg? I'm going to wimp out and go straight to my calculator. I'm going to go 20 to the third power times 10 divided by 16. Oh, if I want to do 1 quarter as an exponent, I've got to put that fraction in brackets. Apparently this is the same as the log base A. Would you all get 40,000? Would you give us one like that on the calculator section? Nope. If you need to go, you need to go? No problem. I'll put all this online and I'll send you, I'll email you all the link. Okay? I hope. It'll be on the website if I don't email you the link, but hopefully I'll email you all the link. 15. Yeah, no, I'm not going to panic. Jordan, what's my base here? That's the log base. Look up, look up, look up. What's my exponential base here? I got a question for you. Jordan, 4 to what power equals 64? Because that's what that equation is saying. 4 to the what equals 64? 4 to the 2? No, what's 4 squared? 16. What's 4 cubed? What's 16 times 4? Yeah. You know what? Now what does that mean? I think, Jordan, that means that this whole mess has to work out to, in other words, here is the equation, I think. The log base 2 of x minus the log base 2 of 5 has to work out to a 3. That looks a little nicer. Are my bases the same? Combine them. Subtracting two logs is the same as? Okay, so if I hear you correctly, Jordan, you're saying the log base 2 of x over 5 equals 3. Where is my x sitting? An exponent? No. Inside a log. Oh, if I know one, I know both. 2 to the third equals x over 5. What is 2 to the third? 8 equals x over 5. Hey, what divided by 5 equals 8 times by 5 if you couldn't see it correctly. What do you do with your x? Dividing by 5, multiplying by 5. Yeah, 40. Here's the base change 1. Number 16. How did I figure that out just by glancing at it? I said, ah, here's the base change 1. What's the base of the if portion? Base 9. What's the base of the then thing, the big ugly one? Base 3. What I'm going to do, rewrite this as base 9. First, let's break it up. What's dividing the same as? So if I hear you, this is going to be log base 3 of x minus log base 3 of 729. I don't want base 3. You know what I want? Base 9. Okay. This is going to be the log base 9 of x over the log base 9 of 3. There's your base change law, right? Minus the log base 9 of 729 all over the log base 9 of 3. Now this looks scary, but wait, is this not on the calculator section? Really? Well then, hey. Oh, first of all, what is the log base 9 of x according to this question? What can I replace that with? So that's a little nicer. 5 over, what's the log base 9 of 3? Now, I've done so many of these, I know it's a half. But if you can't see that, no problem. We're allowed to use our calculators. How would I figure out the log base 9 of 3? Oh, it would go like this, right? Log 3 divided by log 9. That was our base change to find different bases. You do get 0.5, yes? Don't all try it at once. Okay? Right? Jesse, we're good? Okay, this is base change. Any time I give you a weird base like that thing, you can evaluate it on your calculator by going the log of the top thing divided by the log of the base. This is 5 over 0.5 minus, what the heck is the log base 9 of 720? Look, wait, wait, wait. Kyle, I'm not sure I needed to walk through this whole thing. Can I not just figure that out on my calculator? Wouldn't that be a little more efficient? Well, log 729 divided by log 3. Why don't I just figure out the first term on my calculator instead of rewriting this one, Brett, since I'm going on my calculator. Why don't I just crunch this puppy instead? You know what this black thing here, this blue original, you know what this whole thing would work out to? 6. This thing, right? That's going to work out to 6. Or if I type that into my calculator, that also would work out to 6. But Leslie, I went to the simpler expression. This whole thing works out to this. 5 divided by 0.5 minus 6. Oh, I saw you typing away on your calculator. Yes, you. I won't say your name. What are we dividing by here? Dividing by 1.5 is the same as multiplying by what? Are you saying that's a 10? Take away a 6, and you used your calculator. I hope you at least did get 4.00. I hope you at least got the 4. And it's not totally your fault. Your generation sucks at fractions because of the curriculum. It's not a lousy job of preparing you for fractions. Turn the page. We got what? A couple more multiple choice. Oh, a lot more multiple choice, and then we're jumping into the written. Let's see. I may skip a couple of these if they get redundant. 18, this is going to be a lovely quadratic, but it's going to be decimal answers. You know how I can tell it's going to be decimal answers? Because I actually looked at my answers, and they're all decimals. This is going to actually require the quadratic formula to end up solving. And I think I already told you guys that I'm not going to give you quadratic formula questions on your test. If you get a quadratic on your test, it'll factor. If it doesn't factor, Justin, you messed up. Trust me. Trust me. So I'm going to say, ignore that guy. Compound interest. Okay. Final amount equals initial amount c to the t over p. What are they asking me to find in this question? Final amount, initial amount, rate, or time? Final amount, so final amount is on ground level. No logs. This is going to be straight plug-in chug. I'm just going to set the equation up, and I'll let you do the typing. This equals, what's my initial amount then? Oh, 500. What's my growth rate? It's going to be 1 plus 0.15. 1 plus 15%. But here's, we have to tweak it. Compounded how? Quarterly, so divide by 4. What's t time 8 divided by, how long is the period? Quarter. And Jordan, you know that dividing by 1 quarter is the same as multiplying by what? By 4. In fact, that exponent's going to be 32. So it's going to be 500 times bracket 1 plus 0.15 to the 32. Whatever the heck that is. Well, first of all, how much money are we starting out with? After 8 years, there's no way you've got $43,000. If there was, everybody would be doing that. And I doubt C is correct either. There's no way in 8 years that you've not doubled, not tripled, 6-tupled your money. No, almost 10-tupled your money. Anyways, so it's A or B. What'd you get? B? Okay. I don't know how to do pH. Oh, yeah. It was 10 to the bigger minus smaller. 10 to the washing detergent has a pH of 10 minus shampoo has a pH of 8. 10 to the 2, oh, 100 times more alcohol. Just a reminder for you. On your test, I'm either going to ask you a Richter scale, a pH, or a decibel question. Someone finally got it. Number 21, they're asking you to find the percentage growth rate. Nah. 22. Oh, base E, continuous growth. Did they give me the equation? Yep. Which is how you'll know it's base E. Yes, if they use the phrase continuous growth, that's kind of a trigger phrase. But Jen, if I give you an equation with an E in it, then it's base E. If I don't, not base E. Here's my equation where V is the value and T is the time. In what year, I guess they're asking me to solve for T? Will the value drop? Oh, I guess they wanted the final value to be 3,500. Although I got to be careful. They said will it drop below 3,500? It's off to think about how I round this one off. Final amount, initial amount E to the negative 0.028T. What am I going to do first? Yeah, I'm going to divide by 6,930. 35, let's turn the calculator around, 3,500 divided by 6,930. I get 0.505050505050. I'm going to write 0.505, but I'll keep this on my calculator. Where's my variable sitting? What happened to the what? To the 2? Can't you read it? Sorry, I'll make a little larger set for you. I want to get your prescription checked. And a 0? Oh boy. Transcription error, big time. Are you saying I didn't put a 2 up here either? Fair enough. Fine. I'm getting old apparently. Sarah, because the error goes first and then the mind, so I'm a little worried. No bald spot there yet? Okay, gave it time. I'm not going to take the log of both sides. Because this is base E, what's going to work better? Ln. I'll go ln of 0.505 equals, and I'll move the exponent to the front, negative 0.0282t ln of E. Why is that so nice? Because the ln of E is 1, 1. Xt is going to be the ln of 0.505, all divided by negative 0.0282. The ln of that answer divided by negative 0.0282. I get 24.2. Now, if I round this down to 24 years, will it drop below 3500 after 24 years? So I think this one, I'm going to have to round up. I think this one I would say 25 years. Now, we started in 1998, so I'm going to go 1998, and I'm going to add 25 years to that, and I'm getting 2023. And then I say, dope! I guess this question doesn't want me to round up. I guess it's that. I think that's silly. I think it should be 2023. We're going to do one in the homework on the written 23. Well, Nuke, I guess we should practice number 24. It's Richter scale. Remember, Richter scale is 10 to the bigger minus smaller equals the number of times stronger. It says, an earthquake with a magnitude of 5 is 50 times more intense than one of its aftershocks. Determine the magnitude of the aftershock. What are they asking me to find in this question? The bigger, the smaller, or how many times is intense? Yeah? I guess smaller. I think they're saying solve 10 to the 5 minus x equals 50. Good. Solve it. It's brackets, brackets. Because the log of 10 is 1. So I think we're going to have this. I've got to plus this over to this side, e and a minus this to this side. I think x is going to be 5 take away the log of 50. 3.3. 25. It says the number of bacteria in the culture, what, every, what hours? Triples. Every how many hours? How many hours are we talking about here? How many hours? What's the difference between here and here? 10. How many hours to triple? Are you saying that in 10 hours? A little bit. 10. How many hours to triple? Are you saying that in 10 hours? A little more than triple? What's the correct answer without me doing any of the math? It's got to be b. A little more than triple. What does triple mean? What does triple mean? Bob, what does triple mean? 3. So what's a little more than tripling? About 3.9 times as much? Pretty sure it's b. Effective. Right? 8 hours at triples. 10 hours. Probably a little more than triples. Right? Yeah. Sure. Good. Okay. Ignore 26, because that's actually base 2, base 2. I like 27 though. I like 27. 27 is a nice question, like number 27. Leslie, this is what I said you're going to see. Can you see there's a coefficient in front of one of the terms? And then variables in both exponents. That's what you're going to see Wednesday. Right? Friday. Or Monday. Well, when you're at the test. Log both sides. Now, if this didn't have an exponent, if this was just a number, Leslie, we would divide by 5 to get the exponent by itself. But here, if I move the 5 to this side, I still got it in front of an exponent of a matter. Well, I'm going to log both sides. Mr. Dukes says this is where kids make sloppy mistakes. Four o'clock. I haven't changed my, reset my clocks yet. There. The log of 2 to the 3m equals the log of 5, 6 to the m plus 4. And I'd probably make this worth four marks, and that probably would get you a half a mark. Because I think you should be rewarded for cluing in until it's 4 and take a log. Longer than both sides. Ian, the most common mistake kids make now is they move the m plus 4 to the front in brackets. That's wrong. Why? You can. It's the issue here is, if you moved it to the front, Ian, you'd be saying that it was on the 5 as well as the 6. Look, is it on the 5? Otherwise, they would have put an m plus 4 there. I'm going to have to break this one up. What's happening mathematically between the 5 and the bracket? Jen, what is that adding, subtracting, multiplying, or dividing right there? What's multiplying inside a log the same as? I'm going to write this as the log of 2 to the 3m equals the log of 5 plus the log of 6 to the m plus 4. And I can now start moving exponents to the front brackets if they're binomials. And Miguel, I'll clearly see where that m plus 4 is going to end up. It's going to end up here. Not here. Here. It's going to be 3m log 2 equals log 5 plus bracket m plus 4 log 6. Now what? Get rid of brackets. The 3m log 2 is going to drop down like a domino. The log 5 is going to drop down like a domino. And this bracket is going to break into n log 6 plus 4 log 6. That ends on both sides of the equation. Let's get into one side. I think I would choose to minus this guy over to here. Sound effects are optional. 3m log 2 minus m log 6 equals log 5 plus 4 log 6. In terms of what I have on the left-hand side there, I mean what I prefer. It would be wonderful if there was some grade 9 and mathematical operation that I could do. Oh, yeah, yeah, yeah, yeah. GCF. n bracket 3 log 2 minus log 6 equals log 5 plus 4 log 6. Hey, let's take it home. What's happening between the n and the bracket mathematically? So how do I move it over? This is going to be ugly to type, but I'm going to make all of you type this into your calculator because this is your practice. n is going to be log 5 plus 4 log 6, all divided by 3 log 2 minus log 6. Is that right, Mr. Dunick? Check, check, check, check, double check my math. Didn't do anything sloppy. That looks correct. So boys and girls, how's your calculating? I'm going to freeze my screen. Uh-oh. Are you getting a solution? Oh, I tried cheating with my graphing calculator, but I'm 0 for 2, so I'm not doing this the long way. Bracket. Log 5. Close bracket. Plus 4 log 6. Close off the log. Close off the bracket. Divide it by open bracket. 3 log 2. Close off the log. Minus log 6. Close off the log. Close off the bracket. You get 30.5? 1. Now what I was doing while you guys were working is I actually typed this into my graphing calculator. 2 to the power of 3x. 5 times 6 to the power of x plus 4. What did I get for an x value? 30.5, 1. I went to my window, and I went from negative 30 to 30 and wasn't getting a solution because I was estimating it in my head. So I'm going to go from negative 30 to 40. That'll give me a solution. I did backwards. I'm saying I need to clean my windows. How about to 40? Graph. Blah, blah, blah, blah, blah. There's graph 1. You can barely see because it's so steep. There's graph 2. They're crossing somewhere up there. First curve, second curve, guess. Let's see if it finds where they cross this time. Ah, it's having a tough time. Ah! There it is. When x equals 30.5, they cross at 3.55. Oh, no. Sorry. They cross at 3.55 times 10 to the 27th. That's a pretty big y value. They're crossing in the... trillions, quadrillions, quadrillions, quadrillions. I think in the octillions. You had a question? Okay, so there's going to be one absolute lick number 27. I think we're nearly done. Yes? And then I'll take specific questions about specific questions. So here's a good example of a log equation. How do you know it's a log equation? Because where's the x inside of log? Okay. So what's my strategy for a log equation? I want to write it as one thing equals one thing. Are all my bases the same? Okay, then no base change. Brett, what's adding two logs the same as... I'm going to write this as the log base four. Now, look up. Here's how to get a zero. I'm going to get rid of brackets. This is the log base four of six minus the log base four of it. No, no. I've never told you that you can break up adding. Nonsense. Don't do that. And as soon as you do that, I've got to give you a zero because everything else, your pencil just threw up on the page as far as I'm concerned. That's minus x five minus two x equals the log base four of 60, right? Kyle, do I have one thing equals one thing? Do I have one log equals one log? The logs cancel. Now, remember, what if instead of a log, what if I had like a number sitting there by no one and no both? That to that equals that. But this time I had a log sitting there. I should really be careful because we're not canceling the logs. What we're really saying is, well, if you have the log of one thing equals a log of another thing, the one thing equals the other thing. What we're really saying is this, Jordan, what's inside the first log has to equal what's inside the second log. Now what? Foil. Okay. I was going to say get rid of brackets, but even better. Yeah, it's actually, we had the acronym FOIL. You know what? This is weird with the x's at the end. I'm going to show my work. 30 minus 12x minus 5x plus 2x squared equals 60. What kind of an equation is this? How do I know? Make it equal to zero first thing. So I'm going to write this as 2x squared minus 17x. And I'm going to minus the 60 over to this side where there was already a positive 30. I think, Elizabeth, I'm going to have that. Something in front of the square, these are tougher to factor. Yeah, they are. You know the best way to factor in these is, now, I see Troy has pulled out his trusty quadratic solver. I would take that, I would accept that Troy as long as you also wrote down the quadratic equation somewhere right about now. Okay? So if you want to use the quadratic solver, upload it to many of your calculators, sure. But you know what? It would be good for us to review how to factor these just in case. It's going to factor Brett into two brackets. That much I know. Oh, when I'm factoring, what's the first thing that I always, always, always, always, always, always, always, always, always, people at home are going to think my recording is skipping, right? Always, always, always look for. Pat, always check for the greatest common factor first because if that was like a 16, then Jordan I could pull that 2 out and then it's the easy factoring the ones we could do in our hand. Always check for a PCF first. Isn't there one? What's right there, Jordan? The whole thing. I'm pretty sure that came from a 2x and an x. In fact, I know it did. Jordan, what's right here? A negative 30. Now that gives me a lot more options. It could be like a 3 and a 10 or a 10 and a 3 or a 5 and a 6. And when I foil this out, I have to get a negative 17 there. I can't tell you how my brain instantly sees these, but my brain does instantly see that. Check me. See the negative 20? See the positive 3? Will that give you a negative 17 all said and done? Yes, it will. It's the best way to factor them. I honestly let my mind go blank and I just see the answer because I've done so many of these. It takes a while to get there. So quadratic formula, or you can use the box method or factoring by grouping, not groping, just in grouping. There's all sorts of different... I think it's called the decomposition method. There's all sorts of them. Here's my point. Justin, my friend, what are my roots? This one's the easy one. What's the root? Which one? Negative 10 or 10 since you said both? Come on, Bob. There we go. Hey, what's the root here as a fraction? Negative what over what? The limit with using the solver. It doesn't give it to you as a fraction and that might leave you in a hard place. With log equations, we must check for extraneous. Can I go 6 minus 10? No, none of my date people are here. Thankfully, Brett plays basketball. Brett, rejection, no date. What about this one? Well, if I go a positive minus a negative, that's the same as a plus plus. So I don't think that's going to be negative. If I go a positive minus a negative, that's going to be plus plus. I think this one will work. I'd probably type it in just to double-check, but we're in a rush. Number 29. It's okay. Nearly done. Number 29. I think I would say if I gave you one like this, I'd feel more comfortable as a multiple-choice tricky. I'll set this up, but we're not going to solve it. Here's what it says. The number of VCRs sold last year was that many and was decreasing at 18. I'll right underline the word decreasing. 18% per year. As an equation, it would look like this. The final amount. Good question for my ESL students. Video cassette recorders. When I was growing up, we used to watch movies on these tape square thingies, and you could sometimes record over them, but then to fast-forward, you had to press the fast-forward button and you'd hear it go. You'd wait, wait, wait, wait, wait, wait, wait, wait, wait. That little counter would show you how many inches or how many centimeters of paper had gone by, and you'd wait and wait and wait and wait and wait and wait and wait and wait and wait. Those were VCRs. We also had Walkmans when I was growing up, which is not what you think. It was a cassette tape player, which, by the way, just two weeks ago, Sony finally discontinued manufacturing with the tape cassette Walkmans. People were still going with them, especially in Japan. Cassettes, yeah. Ready? Hey, Brett, what's my initial amount? 12. Okay, folks, if I'm decreasing at 18%, what am I putting there? Not 0.18. 1 minus 18%, which in your head is, I heard 82%, which is as a decimal, 0.282, and I'll go to the T, and it's per year, so it's over one. I'm not going to bother writing the over one. Now, that's for VCRs. What about for DVDs? Vice-deVice DVDs? Yeah, okay. That wasn't correct, was it? Well, you understood, good enough. Final amount equals, what's my initial amount? By the way, this test was actually written about six years ago, so back then, back when DVDs were just starting to become popular on the market. Really? What's my initial amount? 7,500. What's my rate? 1.06 to the T. Then it says this, if this pattern continues, how many years will it take until the DVD recorders is equal to the sale of VCRs? See, I think this is what they're saying, Kyle. Solve this. When does the number of VCRs equal the number of DVDs? That's what they're saying. This is a bit, to me, too much work on a written question. So we're just going to talk about how you would solve this. Do I have a coefficient there? Do I have a coefficient there? That's dumb having both. I would divide this one over and write that as one coefficient. In other words, I would go 120,000 divided by 7,500. Oh. You know what? I would solve this equation. 16 times 0.82 to the T equals 1.06 to the T. Now what? Log both sides. In fact, you know what? We're not going to do it. We're not going to argue and suggest to you coefficient, exponent with one base, exponent with a different base. I think mathematically, coefficient, exponent with one base, exponent with a different base, I think those two questions are identical. That's why, to me, this is overkill. We did one like that. Oh, and this is a multiple choice. You know what I would do? I would graph that as y1 with an x right there. I would graph this as y2 with an x right there. And then you know what I would do with those two graphs? Find where they do what? Cross. That's the solution. What if it was on the non-calc? Those are decimals. Would I be able to put that on the non-calc? Nope. So what I do like is number 30. On your written, there is going to be a half-life question. I've done a bunch of these, and there was a couple on your quiz. We did two quizzes that had a whole bunch. Do you want me to do this half-life question? I can. Or, because I've talked for a while, we can just do question and answer and kind of partly wrap it up. That's up to you guys. So if you want me to, I will. But you actually need to speak up and say, Mr. Dewick, please do it. I will. There's no skin off my back. I'm here till 8 p.m. anyways. I got a parent advisory committee meeting tonight at 7. So I'm sticking around. Or if you're kind of going, yeah, you know what? Half-life questions I'm pretty good on. Then I don't want to waste your time, either. So someone would like me to do it. I made the mistake of giving teenagers a choice to be lazy or not. Don't. You know what? Really quickly, I should just write down the answer and that way you can try it on your own. Right? Okay. 0.05 equals 0.5 to the 15 over T. So it's going to be log of 0.05 equals 15 over T. Log of 0.5 15 log of 0.5 divided by the log of 0.5 0.06? I think years, days, years. Half-life is the piece. That's the most common thing. You got to remember, half-life is the period. Okay? Time. Here's how I remember it. What variable, Troy, do we use for time? I remember actually the phrase total time because it's back-to-back keys. And so when I read this, I say 15 years. Is that the total amount of time to get from initial to final? Or is that how long each growth period takes? You know what? 15 years, if I'm reading this question, that's the total amount of times T. And I've memorized that half-life is the period. All right. Is there any specific questions from the review that you would like me to go over that you're wondering about? Yes? Number six. Number six. Oh, number six is nasty. Number six, multiple choice. Have you looked at my solution key, Troy? Let me talk you through it. I'll just talk you through it on here. By the way, folks, as far as I'm concerned, you can keep the tests as far as I'm concerned. If you want to stick around, I can print up the solution key that I just did, but I'm also going to put it online. So it's really up to you. And I know as most of you are writing this down, I'm going to go ahead and do a quick review. Email me. I got more reviews, too. Or actually, wait a minute. Remember, pittmath.com? You click on math 12 links. You click on math 12 links. What does it say right here in big bold letters? Mr. Deweyck loves turtles? I don't think so, but that's okay. Unit one logarithms. I have another unit review right here. I have another unit review that you haven't seen before with solution key. I have another unit review right here. And last year, this is my video tutorial review of the whole unit, except this was at the end of the year. And so it includes log graphs and exponential graphs, which I'm not going to ask you about on this test. In other words, if you're watching that one and I start going off on graphs, just ignore it or fast forward, skip 30 seconds or so. Okay? So that gives you, I don't need to email you the reviews. There they are, right there. All right. Let's answer Troy's question. If you're going, could you put the chair up? Thank you. Troy. Number six. Here's what I said. Okay? Use your pen at the point, Mr. Deweyck, because the microphone has to pick up your voice. Don't stand up. First thing I said is, I noticed right here, my base of the exponent is a and my base of the log is a. My base of my base is, I said, you know what, this whole thing just works up to an x. I dropped that down. And instead of an 8, since I noticed they had a 2 here in my if part, I said I'm going to write that 8 as 2 cubed. So the first thing I did is I rewrote this as the log base a of 2 cubed times x equals 12. No, they want you to solve for it. I'm going to keep going because actually it gets obscure. This is why. I moved the exponent to the front. I got 3 log base a of 2 times x equals 12. What's the log base a of 2 the same as according to this question? So I replaced it with an x and I got ah, well I divided by 3 right now already. I said it's x squared equals 4. I said x equals now initially I wrote plus or minus 2 because when I square root it is plus or minus. But then I made a little note here, x can't be negative if you take the log of a negative so you can see I actually erased the plus or minus after the fact and I said okay, they want me to solve for a. I went back to this expression but instead of an x, you know what I put there? A 2. A squared equals 2. A equals root 2. If you didn't spot that this can't be plus or minus, it's going to be positive. You would have got actually a common answer was D. A lot of people with plus or minus. Oh, by the way on this line a lot of kids say, oh Mr. do it, wait a minute your square rooting isn't it plus or minus? Where is A sitting in the base? Can the base be negative? It's another, so there's plenty of room to accidentally come up with a plus or minus root 2 and it's not. A curve ball will be. Would I consider this fair game, Matt? As a multiple choice as the nasty one down here. Yes. As the written question for four marks or five multiple choice like these. Not a chance. Not a chance. One? Sure. That's why there are some nasty ones on this review. In fact this was from the sample exam questions that they gave us from the ministry of education. So something similar to this made it onto a provincial. Different numbers but the same idea. Any others? Okay ladies and gentlemen I'm signing off. Let me hit stop before I forget. Otherwise this will keep recording.