 I think I heard the magic dings, so I think we're live. You ready? Then shush, in a nice loving teacher kind of a way. No, shush really means shush. Sorry, but shush. I guarantee number one is going to be on your test. Or, instead of giving you an exponent, I'll give you a log and say right up as an exponent. You've got to be able to go from log to exponent to log to exponent back and forth, almost in your sleep. That needs to be a free mark. If you're having to go, oh, if you give me log base j of k equals l and I say what to the power of what equals what and you can't instantly go j to the l equals k, I'll be honest, you're not passing the test. In fact, I'll be even more honest. The odds are pretty good. On Tuesday you'll drop the course. Done this game before. So, absolutely you need to have the log definition memorized. What I've said is if you know one, you know both. If you know one, you know both. So, if they give me an exponent, it's always my base is my base is my base. It's going to be base a, so that's wrong and that's wrong. And which one is it? Okay. Number two says, write this as a single logarithm. Now, this throws a lot of kids off. They go, hey, wait a minute. First of all, there is no rule for dividing logs. You can't cancel the logs. You can't go 12 over 6 is 2. There is no rule for that. However, I noticed my top log. What's the base of the log on the top, Amanda? What's the base of the log on the bottom, Amanda? The fact that they're the same actually means this is the base change law in disguise. Here's what the base change law said. If you have the log base a of b, that's the log base c of b over the log base c of a. This is what the base change law said. So I think what I gave you is same base, same base in the question, yes? I gave you the right-hand side of the base change law. Now, usually, Katie, we were used to starting with the left-hand side of the base change law. For example, Katie, if I said, hey, evaluate log base 7 of 23, you would go log 20 base 10 of 23 over log base 10 of 7. What you were really saying was, by the way, did we ever write the 10s? No, I'll put them there just so you can see that it is this pattern. So I'm going that way, answer log base 6 of 12. By the way, these first nine questions, you should not need a calculator for. Number three, this is a pattern I said that you just wanted to memorize. The pattern is this. If you have a base and then you have an exponent that's a logarithm and it also has the same base as your exponential base, the answer is just whatever that thing is. What's inside the log comes down to ground level and is the answer. Now, it's the same question, Stephanie, as this one. The log base x of x to the whatever, but this one is way easier because Steph, this one, you would all clue in, oh, I moved the bracket to the front and then you would say the log base x of x is one. It would just cancel and you would say, hey, I know this is that. What we're really saying is if your log base and your exponential base are the same, they really cancel each other out because they're inverses of each other and what your term is just becomes the answer. In other words, the answer here is a jude. Number four, a little trickier. What's the y-intercept? Okay, this is a graph. Mr. Dewick, they didn't give me a base. Well, how do I find a y-intercept? It's really math, Dan. How do I find a y-intercept? This always works. Okay? Trevor, behold a graph. A y-intercept is a point somewhere on the y-axis. Every point on the y-axis has something in common. What does it have in common? What? You know how I can find a y-intercept? How can I find a y-intercept? If I ever want a y-intercept, find a y-intercept. Put a zero in for x. I'm going to put a zero right there. I get this. So, I get this. What is zero take away one? Zero? Sorry, what's zero take away one? Negative one? Yes, yes, yes. Did you say zero? I read your lips. I went with it. I should have never even said that out loud. That was dumb of me. You get this, yes? Elevator. Negative one exponent. Really, this first term here is three over a. And the plus three drops down. And as it turns out, that's your y-intercept. Now I'm not going to give you one quite that tough. I would feel free to give you one with a number there. Like an actual base. But you know how I find a y-intercept? Put a zero in for x. That's fair game. It's math down. This one may or may not be on your test. Number five. How can I find the domain without graphing? Ah, by being stubborn and clever. Domain is your x values. What can't I take the log of? Zero is true. I can't take the log of zero. Anything else? There is. Folks, if you don't know, you want to have this at the top of your head as well. Remember all those extraneous roots? Remember Ryan's poor date? What can't I take the log of? Negatives or zeros? Say that again. What can't I take the log of? For a reason. What can't I take the log of, Nicole? That means that I must take the log of a positive number. Let me say that again. That means that I must take the log of a positive number. Let's say that one more time. What's inside the log has to be positive. Can I say that one more time? But I'm going to write it. What's inside the log has to be positive. Which is how you would write positive. It has to be bigger than zero. That's your domain. Except let's get the x by itself. I would, to get rid of that negative 2x. I would plus 2x to both sides. I would divide by two. Three over two has to be bigger than x. Or, if I reverse that, x has to be smaller than three over two. Well, this is still wrong. Really? Yeah. Because there's another x that appears in the equation. Can you see it? Where's the other x that appears in the equation? Sorry? They're both in the log. Any more specific than that? In the base. Yes? In the base. There was a restriction on the base. What can't the base be? Well, we said the base has to be positive. And the base can't be one. And in fact, your complete answer is this one here. D. Now, before you freak out. Very tough. Very tough. In fact, Amanda, I was looking at my notes this morning. And it looks like actually last year, I nuked these questions from the quiz and I printed up my older version of the quiz, not my newer version of the quiz. So if you're going, I have no idea. Well, see, I wouldn't say that you can say I have no idea how to do this. I'm going to ask you probably a domain question for logs. I'm probably not going to give you a base down there. That's tricky. But here's what I will ask you. You can probably write this down on the quiz somewhere or jot this down. I will do something like this. Domain of y equals the log base 3 of 2x minus 4. Nicole, what can't I take the log of? What must the log be then? What's inside the log has to be positive. And then I get the x by itself. I would plus the 4 over. I would treat this like an equal sign. It works the same way, mostly. I would plus the 4 over. I would divide by 2. Domain of this graph, x has to be greater than 2. I equals the log base 5 of 3x plus 6 plus 9. The domain of this graph, 3x plus 6 has to be positive. 3x has to be bigger than negative 6. Domain of that graph, x has to be bigger than negative 2. That's the shortcut for finding a domain when you don't have the graph in front of you. I would draft the log graphs for practice. Here's an easier, shorter way. I do like number 6. How do I find an inverse? The first thing that I would do is I would write x equals the log base 3 of y plus 5. But I'm not done. I need to get the y by itself. Where is the y, Stephanie? Okay, how can I get it out of the bracket? Too slow. How can I get anything out of a log that's on ground level? If I know one, I know both. The first thing I would try is rewrite this as an exponent. What to the power of what equals what? Yes, thank you for getting that. Excellent. Equals what? Equals what? Y plus 5. What's inside the log? Yes? Get the y by itself now. How would you move this plus 5 over, dear? Recover. How would I move this plus 5? See. Okay. I have no problem giving you a log and saying find the inverse or giving you an exponent and say find the inverse. Although if I give you an exponent and say find the inverse, probably the y will end up as an exponent. How will you get it down to ground level? Take the log of both sides and then get the y by itself. That'll be on your test. Number seven, this is also a curveball. Also nasty. Solve for x. Yeah. No. I'm not going to panic. Apparently I know how to do this. I looked at this and I went, well, Tyson, can you read that to me? Log base 3 of 27, no. That's a 3. Isn't the log base 3 of 27 3? I said, you know what? I can evaluate that. That's a 3. I said I now have the log base 4 of the log base x of 3 equals negative 1. Where is the x sitting? Well, it's sitting in two locations. Nicole, it's a base and it's inside a log. Stephanie, if it's inside a log, what do I try doing? If I know one, I know both. I said, you know what? I'm going to write this as an exponent. 4 to the negative 1 equals the log base x of 3. Is that okay? By the way, Nicole, what is 4 to the negative 1 if I get rid of that negative exponent in elevator? Okay, I'm going to do that because I'm running out of room if that's okay. Where is the x sitting? Inside a log in the base, how will I get out of there if I know one and a both? What to the power of what equals what? Amanda, what kind of an exponent is this? It's a fraction, isn't it? In the very first, actually the second lesson of this unit, we learned a trick to deal with fractional exponents. Do you remember? Because you need to for Monday. It's not time is by the reciprocal. It's not multiply each side. I think you're thinking it right, but you're saying it wrong and you're saying it's so wrong I have to shake my head. It had nothing to do with multiplying. It was raising each side to the reciprocal exponent, which is not multiplying. That's a power. Now, when you do that, and this is why I think you were saying multiplying. When you do that, Andrew, it is power to a power, which does mean multiply. If you pick the reciprocal, you'll always have a one on this side. You'll get the X all by itself. Yay. And three to the fourth is three to the fourth is 81. Okay. How many of you got that? I'm impressed. That's nasty. Those of you that didn't. Roxanne, do you have a copy of the quiz? Oh, you did. You were clever enough to go online and get a copy. Good to know one of you is. Thank you. Those of you that didn't, I ask you, did I do anything new in solving that? I also say to you, that's a great example of a curveball. Okay. That would be a good example of a curveball. Number eight, there's absolutely going to be one of these on the non-calc section of your test. Wait a minute, Mr. Duke, I need to use logs to solve this. No, you don't need to use logs to solve this. But Mr. Duke, the X is an exponent. We dealt with X's that were inside exponents before. We knew how to do logarithms, but I had to make sure I gave you a certain type of equation. I had to give you an equation where the bases were the same. Mr. Duke, the bases aren't the same. I see 125 and I see a five. Did I jog your memory? I hope I did. This is going to be five cubed to the two X minus one equals five to the negative one X plus five, where the negative one comes from elevator. Andrew, what's my power to a power rule? I multiply. In fact, I'm going to get this. Five to the six X minus three equals five to the negative X minus five. Nathan, the way I tried to jog this or trigger this into your brain was I always asked the same question. I said, do I have one base equals one base? Yes. Are my bases the same? Yes. Then I can. It wasn't move the exponents down. I can only do that with logarithms. I never used that phrase. Tyson, go for it. Equate the exponents. What I'm really saying is I'm not moving them down to ground level ever. What I am saying is, you know what? If five to something equals five to a different thing, there's something and a different thing. It's got to be the same number. You can't have five to the power of something equals five to the power of a completely different number. What we're really saying is the equation that drops out of here is six X minus three equals negative X minus five. Matthias, now what? More specific how? Plus X, I'll get a seven X. Yes, I agree. And then what would I do with this minus three? Yes, absolutely. It's going to be negative five plus three, negative two. Keep going, Matthias. Now what? C. There is going to be an equation like this on the non-calculator section. It's going to have common bases in disguise. No logs. Number nine. What's the asymptote of this graph? Okay, I don't know. What's my base here? Two. So here's how I told you to remember log graphs. And you can actually memorize the log graph, but Holly, this is my chain of reasoning. I know this is the inverse of that, because base is two, base is two. And Holly, I know that this had an asymptote of Y equals zero. It was the X axis. That means that the log base two has an asymptote of X equals zero, normally. And now I say, how has this graph been moved? And since that's a letter X, Shannon, all I care about is horizontal. How has this graph been moved? It has been moved. Please tell me how. Three left. I'm going to give you either an exponential graph or a log graph or one of each. And I'll either ask you if it's a log graph, the domain, because the range is all real and that's boring, or an asymptote. If it's an exponential graph, Amanda, the range, because the domain is all real, or an asymptote. That's how I'll ask you a graphing question. I was initially going to have you graph a log function, an exponential, like we practiced. I'm not going to do that because I'm worried the test will be too long. I'll just work it into a multiple-choice non-calc question. It's got to be on the non-calc section, because otherwise you could just graph the equation. We have kudos to Andrew who found a typo in here, because that doesn't actually match the question, but that's okay. Let's see. What was your argument? Oh, yes. $4 of interest. He said actually that's 0.004. That's 4 tenths of 1%. He said that should be your rate, Mr. Dewick. Not 0.04. I actually lifted this from an actual textbook, so they had it wrong. Well played. You're saying no. Here's an equation. Then t as a function of v, that's the fancy way Nicole first saying, get the t by itself. So the first thing that I did to get the t by itself was I moved the 1,000 over. I said, you know what? v over 1,000 equals 1.04 to the t. Then you know what I said? I took the log of both sides. Then I got the t by itself by dividing. t equals the log of v over 1,000, all divided by the log of 1.04. Then I went looking for my answer and I went, oh, I don't see it. No, no, no, wait a minute. Mr. Dewick says on the multiple choice section, if you can't find your answer, you're applying log rules like crazy, but correctly. Here's what I've seen some kids do. This is wrong. Oh, we can just cancel out the logs. No. Once again, Nathaniel, there is no rule for, hey, what's dividing two different logs the same as? There is no rule for that. However, there is a rule that if you have two things inside the same log and you're dividing them, what's that the same as? You're subtracting. So I said, I'll try doing this. Oh, don't need the t, Mr. Dewick. I said this is maybe the log of v minus the log of 1,000 all over the log of 1.04. I had no idea where I was going, but I started to smile a little bit. Hey, wait a minute. I see subtraction signs. That's good. I said, ha, that's wrong. Oh, what is the log of 1,000? This is base 10. What is the log of 1,000? I said, you know what? That equals three. I didn't spot that right away. I'm proud I was when I saw it. Then we come to the calculator section here. I guarantee you the first question on the calculator section is going to be something like number 11. I'm going to give you the log of base something of something else that doesn't work out evenly. I'm going to say, do you know how to type this into your calculator? How do I evaluate the log base 8 of 16? Log 16 divided by log 8. Or ln 16 divided by ln 8. No matter what, you get 1.33333333333333333333333. That was the base change log. Example 12. The population of the city grows continuously based on that equation there. Oh, it's Base E. How do I know? Hello, Ee. I see you. Find K if the population grows from 2503,000,000 to 8,000,000 in 12 years. saying is this final equals initial e to the 12k. I could have actually included the millions, 80 million and 25 million, but I think I'll get the same answer with 80 and 25. Where's the k sitting, that exponent? I'm eventually probably going to use logarithms, but not yet. What am I going to do first here? Divide by 25. I get 80 divided by 25, 3.2 equals e to the 12k. Now to get the exponent down to ground level, I'm not going to log both sides. What am I going to do this time because the e is the base? I'm going to ln natural log both sides. ln of 3.2 equals, and the 12k would drop down ln of e. Why is that so nice? Because what is the ln of e? What is the log base e of e? Because that's what ln means, log base e. Keep going Amanda, how to get the k by itself. k is going to be ln of 3.2. You didn't say 0, did you? Oh, sorry. Divide it by 12. Oh, I get it. Shut up Mr. Dick. Yeah, fine Mr. Dick. Yeah, okay. Always poking on the bruise to help you remember the pain so you won't do it on the test. You get 0.06929, 0.097, yes, and positive. By the way, this negative answer is what you would have got if you accidentally put the final population by the initial initial population by the final. I can't remember how I got the other wrong answers. Changing the base. A certain radioactive element has a decay formula given by that. Rewrite it as a base of 0.5. Here's what they're really saying. Take this 1.2 and rewrite it as 0.5 to some power and traditionally we use a letter k there. How will I find what that k is? Shannon, where is the k sitting? Now that you're back here, you with me now? I know. Where's the k sitting? Specifically, how have I described? Exponent. So what will I do to both sides? Shannon, joy of my heart. What do I do as soon as I'm dealing? Huh? What? What? Log both sides, okay? You with me now? You good? Why is that so useful? Because the k can move from here to where? In front of the log I'm really gonna say. We call that Sesame Street says it in front. How do I get the k by itself, Shannon? It's darn right. And I get negative 0.263. So which equation, oh, that's the equation rewritten as a base of 0.5. You just need to put a negative 0.263 in front of the t in the exponent. The written section. Nothing really, guys? I'm seeing heads down. I'm seeing drool coming out. Ryan's yawning. Half-life question. Let's try that again without crossing it out, Mr. Duc. Come back here. Half-life question. A equals a0c to the t over p. Final amount equals initial amount 0.5 to the, and they want the half-life, which means they want me to find p. I'm gonna divide both sides by 400 first. Kirsten, you know what I'm not gonna do? 400 times 0.5. You know why? Because there's an exponent there. You can't be doing that. I'll get 125 divided by 400. I'll get 0.3125. 0.5 to the 25 over p. I'll take the log of both sides and I'll bring the exponent down in front, but Shannon, I'll write it as a proper fraction with two levels, because that way I can clearly see that the p is on the bottom, and I can cross-multiply. In fact, I think I'll get this. p equals 25 log 0.5 divided by the log of 0.3125. You get 14 point. Did I say what to go to in terms of decimals? Now, I'll go 14.9, although if you said 15, I wouldn't freak out, because I didn't tell you what to round off to. 14.9 days. Is that right? If you got that, three out of three. Otherwise, I would probably go, let's see, one, two, three, four, five. There's six steps. I'd probably go a half mark for each step, which means you get a half mark for having that memorized. Yay. Well, ideally, you get the three out of three. Yep, I'm okay with that. The letters are arbitrary. To be honest, I was taught, instead of a C right there, we always put an R right there for it. And I would use that, but your workbook doesn't. I don't want to confuse you. I guess I could have just changed it in the workbook. It also worked as a mnemonic, because then the right-hand side was R over P, which gave you kind of a stupid acronym. We had some, I can't remember some dumb way of remembering it. It's time for, and if then, question. Mr. Dirk, what's a little out of eight doing right there? I reformatted the test, the quiz, to give you more space, and I forgot to change the bottom right corner. Anyways, ignore it. Oh, this is going to be the log of x cubed minus the log of y to the one-half equals three log x minus a half log y equals three times 1.4 minus a half times 2.6. Three times 14 is 4.2 minus 1.3. Is it 2.9? Yeah. There's going to be an if-then kind of a question. There is going to be a Richter scale question. I said to you was 10 to the power of the first earthquake, which I don't know, that's the bigger one, minus the second earthquake, that equals 365 times taller. Holy, where is the x sitting? You know what I'm going to do? What? Log both sides, absolutely. See, that's log both sides. Holy, why is that so nice? What can I now do with that exponent? Move it to the front, and since the exponent's actually a binomial two terms, brackets, yes, yes, yes, yes, yes. I'll get bracket x minus 4.6 log 10 equals the log of 365. Oh, oh, oh, but holy, what is the log of 10? Just time it just cancels, which is nice. I get x minus 4.6 equals log of 365. How would I move this minus 4.6 over? I'm right. I get x equals the log of 365 plus 4.6, which is 7. I'll go 7.16, 7.2, I didn't tie what you're on to. 7.2 on the Richter scale. Number four. Oh, for two marks, half mark, half mark, probably a half mark for moving the exponent to the front and a half mark for the answer. I think that's what I would do. Number four. Oh, population. a equals a0 c to the t over p. Shannon, okay. Oh, didn't want to do that. What's this, 18,000? Really, you're cold and you want to be sleepier? You're putting your jacket on, really? Okay. Sorry, what's this, 18,000? I agree. What's the 7,000? You're losing population, which means my rate better be less than one. Oh yeah, decreasing by 6%, that means 0.94 is your rate. One minus 6%, 94% is what's sticking around. And it's t over, what's the period? What's the growth period here? Since it's annually, that really means every one year. There it is. I'm going to divide by 18,000. The thousands are going to cancel. It's really 7 over 18, which is 3.8 repeating. I'm going to write 3.89, but I'm going to keep this decimal on my calculator, Trevor. 3.89 equals 0.94 to the t. Trevor, my friend, what will I do now? I'd better. And to get the t by itself, I think I'll get the log of 3.89 divided by the log of 0.94. The log of that answer divided by the log of 0.94. And I get 15.26. Oh, it says to the nearest year, so I guess 15. Hey, 15.26 equals 15 years. If you didn't round off, I'm not going to freak out. If you rounded off wrong, if you said this was 15.2, then I'll take a half mark off. It's 15.3, or 15.26, or 15, 0.389. Yeah, I think I did it right on my calculator anyways, which is why I use the answer button whenever possible because I'm stupid that way. No comment, Shannon. You were going to say, oh, I only got an I and an O mixed up, Mr. Dewey. You got a decimal and the wrong F on it. I agree. Is that okay? Kirsten, we're good? Three marks, probably one, two, three, four, five, six steps, half mark for each step. Then the last couple of pages of your test are going to be some equations. How can I solve this? Stephanie, where is the X sitting? You know what I do when the X is either in the base or in the log? If I know one, I know both. That's my first strategy. X to the negative three over two equals 64. What's on the X now, an exponent? What type of exponent? Y. It's a fractional exponent. I have a great trick to deal with a fractional exponent. What, Oprichel? What's the reciprocal of negative three over two? Got to be fussy. Yeah. Okay. Why is that so nice? Yay. Those end up becoming a one. I end up with the X by itself, X equals 64 to the negative two thirds, but I'm not done. Oh, negative exponent, elevator. This is one over 64 to the positive two thirds. Oh, fractional exponent, flower power. Now you could go to your calculator, but just in case this showed up, hint on the non-calc section, hint. This is going to be one over the cube root of 64 squared. Amanda in the front row, what is the cube root of 64? Squared with authority, like you know what you're doing. You know what the bottom works out to? And there's a one on top. In fact, the answer X equals one over 60. In marking this, if you rewrote it as an exponent, I would give you a half mark. If I saw that you'd put both sides to their reciprocal power, I'd give you a half mark and then one mark for the answer. Absolutely B is going to be on your test, or I can go a little bit tougher than this. The one that's a little bit tougher than this was when we also stuck a coefficient in front. I didn't put one like this on the quiz. I'm wishing I had, but it's okay. What am I going to do? Log both sides. Oh, am I going to take any shortcuts here? No, no. You'll notice I've taken shortcuts Kirsten for the population questions. I've taken the log and moved the exponent to the front in the same line, but the reason was the exponent was almost always the only exponent and it was a nice monomial with no other junk. Two exponents, binomials, no shortcuts here. Log both sides. Exponents to the front with brackets. Get rid of brackets. Get the X's to the same side and get the non-X's to the other side. I think I'm going to plus this X to there and I'm going to minus this three log seven to there. Andrew, how many X terms do I have? How many would I prefer? It'd be wonderful if there was some kind of a grade nine mathematical operation. I'm trying to give you a bit of a trigger phrase for your brain to go in that direction and you all now say, oh, I can factor the X out, but I got to move over. I would get log two minus three log seven equals X bracket for log seven minus plus, that's what I said, log two. Can't make a plus sign without doing a minus sign first. Give me time. That's true. Can't do that. Okay, a man, the little miss, I'm so smart. How do I get the X by itself? How do I move the bracket over a minute? X is going to be log two minus three log seven all over for log seven plus log two. Now I go to my calculator. Is there more than one term on top? Brackets around the top. Is there more than one term in the bottom? Brackets surrounding the bottom and then close off each brackets for your logs that you're going along to. It's going to look something like this. Bracket log two minus three log seven, close bracket, close bracket divided by bracket four log seven, close bracket plus log two, close bracket, close bracket. Pretty sure the answer is negative 0.607. Is that right? If you got that and you showed work, full marks. By the way, one, two, three, four, five, six, probably a half mark for each of those and then one mark for the answer. That's how I'd make it out of four. Nearly done. I'm getting close. What type of equation is this? It's a log equation. You know what? Get all the log stuff to one side and all the non-log stuff to the other side. Now if you have logs in everything, you would just combine the logs, combine the logs, re-read it as one log equals one log and then the logs can't. No, they don't. They take the analog both. They cancel. Here, since I don't have logs in everything, I'm going to plus the four over and minus this over. I'm going to write this as the log. Who cares? It's a base. I made it look scary. Base root x of x minus two minus the log base, sorry, root x. I said root two, base root two of x plus one equals four. Tyson, what's the base of the first logarithm? How about second logarithm? They have the same base? Then I can combine them. What's subtracting two logs the same as? This is really the same as the log base root two of x minus two over x plus one equals four. Do I have one thing equals one thing? Yes. Do I have one log equals one log? If I did, then the log. Holy, where is the x sitting? Oh, if I know one, I know both. I'm going to rewrite this as an exponent. What to the power of what equals what? Well, I think it's going to be root two to the fourth equals x minus two all over x plus one, right? And as it turns out, root two to the fourth power actually, actually it works out evenly. I think you, or yes. No, yes, yes, yeah. Root two to the fourth is just, I get this, four equals x minus two all over x plus one. And it's amazing how many kids panic here when actually this is the easiest part. This is map eight. You know what I'm going to do? This is cross multiply. I see kids cancel x. I see all sorts of stuff. No. I thought I'd made this easier when I gave this. It's cross multiply. I'm going to get four bracket x plus one equals x minus two. I'll get four x plus four equals x minus two. I'll get three x equals negative six. And I'll get x equals negative two. Not done yet. Not done yet. Ryan has only phoned her up. She hasn't actually said she'll go on a date. Let's go check if it's rejection or not. To do that, I plug it in here. What is negative two take away two? Can you take the log of a negative, Nicole? Oh, Ryan. In fact, you know what? No date for you. Sorry. Although, look, I mean, ladies, he's wearing the tie and the dress shirt. He's working it. He's working it. We have hope. We have hope. And again, you're one in the long line of basketball players for the past 10 years of whom I've made fun of their dating life for the rejection. How many got that? Okay. I added these last couple because I like some of them. Not all three of them, but I kind of like number six. Okay. Here's an if then question again. All right. If what this? Let's run through my usual checklist. Are my bases the same? Yeah. So I'm not going to pull out the base change law. Did they give me one log or several logs in the then section? One, I'll break it up. If they gave me several, I'd try combining it. I'm going to break this up now. I'm going to break this up intelligently. What numbers right here, Amanda? What numbers right here, Amanda? What numbers right here, Amanda? Not 15. I don't see that. You know what I see? I don't see a five either. Ready, Cassandra? What numbers right here? What numbers right here? That's not a 15. You know what I see? Nicole, louder. I see five times three. Do you see how there is that kind of hint there? I see that. Why do I see that? Well, first of all, what's multiplying two things the same as? This is really the same as the log of five plus the log of three, oh, base two, base two minus the log base two of two because dividing is the same as subtracting. This last term is nice. What is the log base two of two? Amanda? One. Oh, and according to this question, what's the log base two of five? What can it be replaced with? And according to this question, what's the log base two of three? Because I asked you to, what's the log base two of, I will never get tired of that joke. Turns out x plus y minus one. So if they had given me, for example, I don't know, a 30 right here and this had been a three and a 10, I would go, oh, that's 10 times three. If they had given me a 60, well, if they had given me this right now as a 30 right now with these numbers, I would go, oh, it's five times three times two and two is good because that's my base. I stuck one more of these on here because kids have trouble with it because I like this question. I think you're going to see one of these on the multiple choice as a nasty. A log inside of a log inside of a log, which we did already once before. This one's a bit trickier. It's an equation inside. Well, you know what? I think the first thing that I would do is I would rewrite this. Now, the one that we did on the multiple choice, I could evaluate this. So I started on the inside and worked my way out here because I can't evaluate this. I'm going to start on the outside and work my way in. Two to the power of one equals the log base x of x plus six. You okay with that, Steph? By the way, what is two to the one? Two. Oh, that to that equals what's inside the log. x squared equals what's inside the log. Is that okay, Amanda? Yeah. And now, what kind of an equation is this? It's quadratic. How do I know? It's got squared. How am I going to solve it? First thing before I do anything else. I get equal to zero. Will it factor? Let's find out. If it factors, I'm looking for numbers that multiply to negative six and add to negative one. Are there numbers that multiply to negative six and add to negative one? x plus two, x minus three. What are the roots? Ryan's just following up two girls. We need to see if there's a rejection. Now, when I put both of those into there, I'm good. But I do have to reject this one here. Do you guys know why I have to reject that one there? It's not because of the x that's up there. Base can't be negative. The base can't be negative either. But the good news is Ryan gets one date. Last one. How long would it take? Okay. It's compound interest, which is based on this. Final amount, a million dollars. Initial amount, 12,000. And then here's where we had to make our little adjustments. C is going to be one plus, oh, not 7.5, Mr. DeWitt. Nice try. 0.075 divided by what? Four, because it's compounded quarterly. And it's going to be T over one quarter. Your period's one quarter. By the way, dividing by one quarter, Katie is the same as multiplying by, dividing by one quarter, same as multiplying by what? Four. On my next line, I'm going to write 14, because we're less than two million. Okay. And you know what else I'm going to do? I'm going to go one plus 0.075 divided by four. From now on, inside the brackets, Katie, I'm going to write 1.01875, which isn't too much right. Okay. What am I going to do first? Yeah, I agree. One, one, two, three, four, five, six, million divided by one. I'm going to write 83.33, but I'll keep the 83.3 repeating on my calculator. I get 83.33 equals 1.01875 to the four T. Now what? Log both sides. Get the T by itself, divide, divide. The log of this thing divided by, oh, I got more than one thing in the bottom. I better use brackets. Bracket, four, log 1.01875. Close off the log. Close off the bottom. And I get 59.5 years. I would probably give you a half mark, half mark, half, let's see, one, two, three, four, five, six. Yeah, a half mark for each little step here. Give your, there it is. I hope you found that useful. Give yourself a score out of 45.