 So what we're gonna do then is we'll do this as a practice test. I won't do every question sometimes I'll say are we okay on doing this and everybody nods their heads. I'll skip that question, but let's start here now Your test is gonna have a non calculator section. It's gonna be nine questions multiple choice Okay, what will they be? I bet you lots of log rule stuff in different bases where if you had your calculator You could just type it in but I'm gonna force you to evaluate it using log rules For example stuff like don't write this or you can write this down stuff like this for example evaluate the log base to That's not gonna work. Mr. Do it Evaluate the log base two of your different base. Let's try that again evaluate the log base five of 250 Minus the log base five of 10 that would be a good non-calc question as a choice question Because with a calculator you would go base change base change and just type in it. How would you do this by hand? What's subtracting two logs the same as? Yeah, this is really the log base five of 250 over 10 and what is 250 over 10 I Had expect you all to be able to divide by 10 without a calculator And so you're saying Alex this ends up being the log base five of 25 Oh, what is the log base five of 25 to that'd be a great example, although I'll probably also do some algebraic ones In fact, you know what I think it says here complete questions one to ten without a calculator. So let's try these How would you solve number one? Yes? What did I say? It's nine multiple choice questions. Okay? What it's gonna happen is I'll start out the test by saying everybody put your calculators on the ground You all will I'll hand out part one Nine questions. I'll say you should budget about 20 25 minutes Unfortunately, you're not gonna be able to go back and double-check your answers So make sure you double-check your answers before I give it to me when you're done part one You're gonna walk up here hand it to me. I'll give you part two and I'll say go pick up your calculator When you're done's part two, you'll come to me I'll stable part one on the top so that they're together and that's how it'll work. Okay, how would you solve this? First question I ask is where is the X is it an exponent? Then if it's in the log I always use if I know want to know both I Write this as an exponent What to the power of what equals what? X to the one half equals 10 Okay. Oh, oh, we also had a something that would let us get rid of exponential by the way Please don't write in these. Please don't write on these. Please don't write on these. Please don't write on these Please paper Thank you We had if we had a fractional exponent. We had a little trick. What was the trick for a fractional exponent? Trevor Both sides don't forget both sides Why is that so nice here? I get X equals 10 squared Trevor. What's the correct answer here? ABC or D? Thank you Okay, I did say I expect you to memorize certain exponents and by now you kind of get an idea what I meant by memorize You gotta be able to recognize number two I think this would be a little over. I this is great on the non calc except I don't know if I'd have three so this is a little bit tougher than you'll probably find which is good practice What number is that right there Emily? No, it's not. What number is that right there? It's not a nine. What number is that right there? It's not an 81 because this is non calc. What are all of those actually Emily? Threes okay, if it was on the calculator section, well, maybe I could log both log all three side You know what this is non calc. That means we're gonna do the old write it as are the bases the same That one basically something that I can equate the exponents, so I'm gonna write this 27 as a Three cubed To the 3n minus one Times I'm gonna write this one ninth, but three to the what? Negative two. Yes elevator. I'm gonna write this 81 is three to the fourth To the n plus two what am I gonna do with these two exponents here? Okay, so I'll get three to the four n plus eight What am I gonna do with these exponents right here still the same yeah also multiply them I'm gonna get three to the nine and minus three here. There isn't a power to a power so the times Three to the negative two drops down like a dominant It's so you look in a little puzzle is that okay so far enough because I can't write 27 as a nine It's nine times three, but that's not an exponent nine to what power equals 27 Right and and also by habit I almost always go to the smallest exponent by default anyways, but not yet nine Doesn't write 27 as an exponent. Is that okay now question for you What's that right there times What do I do with these exponents then? Yeah, this is why I Mean I kind of like this. It's a nice little bit of simplifying here Katie. I'm gonna get this Three to the power up is gonna be nine n minus three plus negative two It's gonna be three to the power of nine and minus five. I think equals three To the four and plus eight But yes, do I have one base equals one base or my base is the same? Then I can equate the exponents my equation is actually going to be and I'm not saying the exponents cancel I'm saying look three to something equals three to something the something's got to be the same thing My equation is gonna be nine and minus five equals four and plus eight Now what more specific than solve for n because that's what we started doing right here actually More specific how here I would probably go Minus four and from both sides and it's all at the same time because I like to do two steps that once because I'm lazy I would plus five to both sides. I think I think I think and I'll get this Five and equals 13. It's all what is n equal to? What's the correct ABC or D? Okay There is going to be some kind of a common base solve exponential equation on the non-calc section of your test. I Just don't think it'll have three terms So oh This is an if then question You know how I know because I see the word if now they've worded this backwards from the way I worded to you I always said if this then this but this is the them Okay Let's see what we can do here. What's my base in the if Emily M. What's my base in the if and the base is M. Really? And what's my base in the then? M M. Yes, it is. So am I gonna use base change? No, so my next second strategy is yep No, sorry. What? Here's what I said you always want to change the then so if you phrase this as if something then what would you not Say if this then what's that always focus on the then not the if I'm gonna be mucking around with this thing Okay, but I started out because I'm trying I'm trying Brett to turn this into this first thing I always check is are my bases the same because if they weren't that's screaming out to me Hey, try changing the base of this into that But they're both what M Base what M? Okay. Now Brett I have one term I'll try breaking it apart if I have more than one log. I try combining it So I have an n squared divided by M. How do I break that apart? What's dividing two things inside the same log the same as? This is gonna be the log base M of n squared Minus the log base M of M. Oh This I like what is the log base M of M? This is gonna be something minus one any suggestions here. Yeah, why not so that would give me two log base M of n oh That's handy because what did they tell me in my if statement that I can replace every log base M of n with a what? Apparently this here is a six So this is really two times six minus one and I think that's fair game to expect you to do without a calculator What's the correct answer here, I think it's be although my one has been chopped off. I'm pretty sure it's be Is that where the whole punch is or something like that? Okay Number four Now this is also sort of an if then question except all they've given me as an if They haven't given me a then to work with instead. They've said write an expression for what? Write an expression for what? Means get the a by itself Okay Hmm Well the a is here before I get the a by itself. You know what I should get by itself the log a I'm gonna move this one over I'm gonna have to minus it because in front of it is a plus sign. I'll get this negative log a equals log b minus five log c Minus one. I want to get the log a by itself. What's right there? Shannon Shannon You know what? positive negative positive divide everything by negative one that okay so far Brett What are we trying to get by itself? I? Think I should try and write this as one term now. Here's the problem That's a log That's a log What's that? More specific More specific a number what? Okay, you ready scene one act one and take two. That's a log. That's a log. What's that? No, it's not and I that's why I wanted you to say that I sorry it is But I don't see that you know what I see Isn't that a one? Because I want logs and everything to combine things don't I I'm gonna rewrite this the log a is gonna drop down negative log b Plus and I'm gonna move the exponent up there because we said we want if we're gonna combine things We don't want coefficients. We want them as exponents and I'm gonna say yes That's the one but you know what I can write any number. I want to as a log What if it had been a two? It's a log of a hundred Seeing as we're based 10 if it been a different base, I would have put a different number whatever the base number there was Okay Now I can combine this log a equals Are these all three the same base? Yeah, do I have any coefficients in front? Nope, and then when I combine this I gave you an easy way to remember if it was complicated. I'd said positives on top negatives on the This is going to be the log of There's gonna be a C to the fifth on top There's gonna be a 10 on the top can I put it in front because that's where we're used to seeing it And there's gonna be a B on the bottom Is that okay trap? Okay You know what I have now One log equals one log Yep, where here? Here Where'd it go right there? Positives on the top negatives on the bottom. I elevated it buddy. I elevated it Is that all right By the way, Alex, do I have one log equals one log then the log can't okay? We're taking the anti log shut up. Mr. Do it. They cancel My next step I can say look if I have the log of something equals the log of something the something has to equal The something by the way, what did we say this question wanted us to get by itself? Ta-da Now let me paw about correct answer is a Did I do anything new there? I tried to tell you how my brain would approach this I said well if they want the a by itself. Let's get the log a by itself Move the one over how as a plus sign in front of it minus it move the negative over how divide everything by negative one Then I said well since I have a single log over here. I'd like a single log over here Hey, I can write a one as a log. That's a handy trick by the way You can also write a two as a log or a three as a log and write any number as a log if you need to Combine my law move the Coefficients back up to exponents and then negatives on top buzz out of the bottom. There you go That's a tricky question number five which of the following is equivalent to that I Glance at my answers. I think it's obvious. They want me to combine the logs You know how I know How many logs do I have here? How many do I have in all my answers? Yeah, are we learning hopefully on multiple choice to glance at the answers for hints because it really helps especially math Okay, I'm gonna use my log rules. Oh What's dividing by two really the same as multiplying by I When I see that I also automatically see this Now why is that nice? What can I do with that one half by thinking of that too as a one half like that? What can I do with that one half? Amanda Because if I'm gonna combine them I need to make them exponents again. I can rewrite this as the log of a to the one half plus The log of B to the one half. What's my base here? What's my base here? That doesn't look like the word log very good. Let's try that again there look like a move base 10 base 10 Do I have any coefficients anymore that I can combine these what's adding two logs the same as I don't see that yet? Oh, oh, well, mr. Dukes says start using your exponent your log rules until you do see the answer What do you see in two of your answers here? Amanda square root. Oh, what's one half as an exponent the same as? Oh, so you're saying this is rude a root B Now the real question then the real question here M is What does that simplify to does rude a root B simplify to root a times B root a plus B? Can I say those are wrong? I'm pretty sure Yes, is root a times root B root a B now this is math 10. They're sneaking it in How do you multiply square roots? If I gave you this Three root six times five root 10 How do you multiply square roots? The answer is it's numbers by numbers Roots by roots and That's actually how I remember it's numbers by numbers roots by roots if there were no numbers there it's roots by roots six times 10 is 60 It is a times B. It is That because how do you write times something? Don't you just write them side by side? root six times root 10 is root 60 root a times root B is root a B Sure Yes, but the real question is okay. What about it with exponents the fact that the only reason we can do that and Multiply and go further is because the exponents are identical if it's a bit of two-thirds and a one-half We're stuck with eight of the two-thirds times B to the one-half We can't take it any further you can't multiply cube roots by square roots or fifth roots by fourth roots There's no easy way to do that Number six says which of the following does not have the same asymptote as that. Okay. Here's an asymptote question What does every first of all is this an exponential graph or a log graph? Exponential all of you Hopefully by now it's ingrained. What does every single exponential graph look like all of you? What does every single okay, by the way, this is why I did that stupid thing every time I figure if I get one person do it you guys remember so what's every single exponential graph look like or? The real question is what does it have as its asymptote originally? The y-axis sorry the x-axis mr. Which as an equation is what so any exponential function Asymptote is the x Axis what's the equation whoever said greater than that's a domain or a range? I want the equation of the x-axis Okay, and seeing as there's this is one you hint hint wink wink nudge nudge. I like this question I'm gonna ask you about asymptotes almost certainly It's original asymptote the equation of the x-axis is a line zero high and On a graph. What's another word for height? Why? That's the equation of the x-axis. That's your normal exponential graph How has this graph been moved? Okay, which of the following does not have the same asymptote as that? What's the asymptote of this graph? Is it an exponential graph? How do you know? Where is the x-sitting? So is it an exponential graph? How has it been moved? It's got the same asymptote and they want the one that doesn't have the same asymptote Is this an exponential graph? How do you know? Where is the x-sitting? So is it an exponential graph? How has it been moved one left and? Will one left change this at all It has the same asymptote Okay, now These two here you'll notice they switch the x and y around These are inverses of each other. I'm not going to ask you one this tough. I don't mind talking about it right now I would have given you two more exponential graphs to pick from instead of giving you But anyways What did you say this had been moved? Alex Which one of these has also been moved five down? Which one has to have a different asymptote then? This has been moved five up I don't like this part where they instead of having a y there in an x they've given you an exponential graph in logarithmic form Do that forget it. I'm not going to give you one that yucky But I would say I expect to be able to give you this and say tell me the asymptote You know what I'd have for my answers x equals five x equals negative five y equals five y equals negative five Or If I gave you something like this x equals negative one x equals positive one x y equals negative five y equals positive five That's a more realistic. Hey, there's four multiple choice ones to pick from number seven Find the x intercept of that graph there Look up I'm not going to worry about x and y intercepts. Although you should be able to find them. That's math 10 However, I will feel comfortable asking you a domain question number eight Ah, nothing I got Brett back there. She's having trouble you're gonna make it because she's actually doing this trev log question or exponential equation Okay Domain what can't I take the log of? there are technically two things or The one you guys always forget zero. Okay, and what we are saying let's reverse that what must I take the log of What's inside the log? What's inside the log? What's inside the log? Has to be bigger than zero positive Amanda what? I think you were right Oh, well get what what I do now Plus the 10 over Then divide by five Okay, you we initially found the domain by actually graphing these By saying hey, this is base 10 So it's going to be uh one 10 to 100 and we found the inverse and then we did our slides But what there is a shorter way if all you're interested in is the domain You simply say what's inside the log has to be bigger than zero done I'll email this out as a pdf as well. I think okay uh Yeah, I will It's copy right, but this company's not going to publish these anymore after this year That's fine number nine Turn the page Did I circle it correct? Yeah, okay Now number nine Although they've done this as a multiple choice I'll probably give you a question like this as a written. This is a log equation You know how I know it's a log equation because there's an equal sign with logs See it This question isn't actually asking you to solve It's saying When you simplify what equation will you end up with? Okay What do we say our strategy was when we're dealing with the logarithmic equation? What do we want to write this as? one thing I'm going to say one thing equals one thing either one log equals one log logs cancel or one log equals one thing I'm going to combine the left hand side because the right hand side is one thing to three Yes, uh I can't combine them yet. Although subtracting does mean divide. That's a problem Although what can I do with that two? Are my bases the same? Yep. I can write this as the log base two of N plus one on the top positives on top Negatives on the bottom and the three drops down Now it's so if I had log base two equals log base two then the logs would cancel Do I have one log equals one log? No. Where is my end sitting? Inside a if I know one or both Right that was our strategy. Basically if it's an exponent log both sides If it's inside the log either in the base or in here if I know one or no both I'm going to write this as an exponent What to the power of what equals what? What's the correct answer? Which of the following equations will result? Ah, you can see what they're trying to trick us with right? They're wondering which kids cancel the logs Because they would just keep a three there That's wrong. That's wrong And oh, they're saying some people will actually multiply the two instead of moving it as an exponent. They'll like foil it or something like that That's wrong Hey, is that okay? Now for the record if you did have this hint hint wink wink nudge nudge What would you do now to solve this? fast, okay cross multiply foil this out Get a quadratic and probably a fairly ugly one I'm not going to give you one this ugly But I am going to actually expect you to be able to handle something with ends on top ends on the bottom Or x is on top x is on the bottom and a number over here. You better not freeze up cross multiply I've done my big song and dance about it, right? This would probably be a this this will be a written question. No I'll give you a log equation on the written fact. I'm going to give you two count them two log equations on the written section One will probably end up with a quadratic Cool one won't What else did you have to remember when you were solving log equations? Someone's date Right I have to check for extraneous roots Okay, number 10 two students solved this using different methods Their work is shown below Whose solution is correct? I guess what I'm really doing is looking to see if anybody made a mistake So let's follow jen's first Again, I'm not going to give you one like this. This is too much typing. I'm too lazy for a multiple choice question But this is good practice Where'd the fours come from? Sorry. Oh, I thought you're right. Where'd the fours come from? Okay, is that correct is 16 four squared and is 64 4 q. What would you do with these exponents? Okay Is that right? why In fact, they didn't go two times three. What were that eight they went two to the third didn't they? So that's the mistake. So jen's is wrong Which means it ain't both Internet jens It's either bobs or no ones now. Let's look at bobs method Bob didn't rewrite this as an exponent. What did he choose to use instead? Log both sides. Yes Now what can you do move that to the front? Now what can you do Move it to the front. You can't cancel the log Whoever's saying cancel the log. No never. Uh, I gotta give you zero if you start doing that Okay Is this correct? Yes, it is Is this correct? It's a what? Brackets binomial. It should be this should be in brackets because you can see on the next line all of a sudden instead of Multiplying the log and expanding it. I don't know. He's got an extra I don't know what the heck happened here. That's a mistake. You know what neither is right Jan is that okay? Okay, that gives you a few ideas of ways they can ask non-calc type questions Okay, I'm definitely going to ask you to simplify some logs I will on your non-calc give you a word problem or two either exponential growth Half-life base e Compound interest Richter scale and instead of having you solve it I may just say which of the following four equations is the correct equation In other words, can I test have you memorized that a equals a zero c the t or you can use it That's a nice way for me to test. Can you even fill a stuff in before solving it? um Definitely going to be an if then question on the non-calc section um That's about it Then we get to the calc section Okay Don't freak out about 11 Number 12. I'm not going to say don't freak out about in that there is part of this that's useful You know how I would solve this On the calculator section Look at your answers. Do any of them work out evenly? If they're decimals that's a hint that you're supposed to graph this y1 graph this y2 And find the point of intersection where they cross I am not going to ask you that on the test But did I teach you how to check your answers on the written section for exponential equations using that method? Well, then that's handy but No 13 is good though Which of the following is equivalent to that? What don't you see in any one of your answers? Okay So apparently there's some way to make the logs vanish Suggestions Let's try moving the exponent to the front You know what I wonder Vlad I wonder if the log base 8 of 512 actually works out evenly evaluate that on your calculator Which is why this is on the calc section And I didn't have you memorize 8 to what power equals 512 which is why it's on the calc section Now how do you evaluate log base 8 of 512? What do you type in? Anyone log 512 closed bracket divided by Log 8 closed bracket. What is The log base 8 of 512 is it 3? What's that simplify to and don't you dare say 7m? 12m Too often too often Man that we do the tough stuff and we're so thrilled with being done the hard part our brain relaxes You'd be amazed how often when I mark someone's test especially in physics the mistake is made on the last line And it's a dumb one right because they've got all the hard stuff. Ah, I'm home free. I like number 14. Let's see Try number 14 on your own Try number 14 on your own I'll do it up here You know you have it right in front of you Joel. I know you're looking up and copying it, but it's right in front of you Just be thinking is that right Since this was on the calculator section Brett although I know 16 to the one quarter is the 4 through to 16 Which is to I forget it. I'm like, you know what it's this on the calc section Do I want to save time? Do I want to make sure I don't make a dumb mistake as soon as I got to here? I just type that into my calculator as is 20 to the third times 10 divided by 16 to the power of oh There's more than one number in the exponent one over four. I better make that a fraction uh brackets You look in the funnel bretter. Is that okay? Oh, would you like to borrow a graphing calculator for the duration of this tutorial? Oh, yes your batteries girl, aren't you? I'll give you my silver so you won't accidentally walk off with it If you don't get batteries this weekend bring your graphing calculator to the test and I'll be willing to swap But I'll also make fun of you. I don't think it goes in that way get him There you go Is that all right? Joel, okay now I would have no problem if these numbers it's all we're smaller This would be a great non-calc question too except I'm not expecting you to be able to do 20 to the third power in your head But is it fair game for me to expect you to do 10 in your head? Yeah, I'm multiplying by 10. Is it fair game for me to expect you to go four through the 16? I think so Where'd the fourth root come from trevor one quarter exponent is fourth root, okay? number 15 suggestions here I'm not gonna panic apparently they think I can handle this Emily, what's your base here? Can you say it with authority like you know what's your base here? What's your base here? It's not a 64. What's your base here? For what's your base here? So 64 is what really? First thing I think I would do is I would say I'm gonna do that I'm gonna do this Because Vlad you know what I can do now Do I have one base equals one base? Yes. Are my bases the same? And you know what I can do with those exponents Wait them now. How did I spot that? You want to be alert? I mean if they give you a four and a 64 that should hopefully by now Kind of be screaming out at you a little bit or if they give you a five and a 25 Or a three and a nine, but I'm not in the 27 There should be some screaming out at you Okay for me Then you can't solve this The way we're going so you'd have to go graphing calculator or something But do you notice your answers aren't decimals? That's why I figured we'd probably do both this by hand because now what's the actual equation we're going to solve this? The log base two of x minus the log base two of five equals three there's equating my exponents Now this is a log equation. I want to write this as one thing equals one thing Are my bases the same here? What's subtracting the same as? Oh, so this is the same as the log base two of x over five Joel did logs cancel? Really? I can just go like this Really? Why not because to do that what what I need over here? But where is my x sitting location wise is it an exponent? Where is it then? In Okay, we have two strategies That's an exponent log both sides if it's inside the log if I don't want to know both Right So what to the what equals what? x over five equals two to the third How would I get the x by the way two to the third is what? Eight not six right eight. Um, how would I get the x by itself? What's the five doing right now? Dividing so I want to move it over You know what I think x is going to be five times Eight I'm going to say that would be fair game as a nasty non-calc I'm not saying at it, but I'm just trying to give you an idea of what I consider easy what I consider tough What I consider technically you don't need the technology for what do I say you do need technology for turn the page Brett here's another if then question You know how I can tell if Except once again the author of this has phrased it different from the way I do the if is the log base nine of x equals five The then is the mess I'm going to work with This boy Something is screaming out to me right now just hollering at me with all of its might right now. There's something just shouting to me What am I going to try Why base change Base what? three base what? And are three and nine nicely related though. Ah, so good stuff's going to happen I'm going to rewrite the then as base nine It's going to look like this log base nine log base nine x over 729 three That's the base change law right? Your original base is in the bottom Your original log is on the top and it's the new base and the new base Ah, just thought of a dumb way to remember your original base goes to the basement. Ah, I'll use that No, never mind. I'm not teaching that well next year. Okay Thought of a great way the final time I'm teaching this wow Where was that for the last 10 years the base goes to the basement that's brilliant mr. Do it Stupid how tough was that to come up with 10 years ago? Now is this calculator section or not? Yeah Then what's that? Can you type? I don't think it's three. What is the log base nine? What is it? By the way, I could try and do this all by hand if it was non calc I would I guess I would but are we on the calc? Let's turn our brain off and It's going to be log three over log nine. I think 0.5. Yes. Yes. Yes. Yes. Yes. Okay What can I do with this? Separate it, you know why because I think that's going to give me a log base nine of x Which I can replace with up. Okay, so if I separate this I'm going to get this the log Base nine of x minus y minus dividing but right is a log Base nine of seven hundred and twenty nine And I know that I can replace that with a five Here's a point five What's this? Is this non calc? No. Well then what's this? Evaluate it. What is it log base nine of seven hundred and twenty nine? I think four Three really? Oh, yeah, because nine squared is 81 81 times 10 Is 810 81 times nine to be close to 800. Sure Okay Now again, you know what I'll say if these exponents were smaller I would feel comfy with this as a nasty non calc, but it's on the calc section What's five take away three? divided by 0.5 What was that? I said for graphing. I'm going to ask you either an asymptote or a domain range Number 10 to me is partly fair game. First of all, I glanced at the answers. What do all the answers have in common? Decimals, you know what if I give you this it'll kind of be on the written What type of an equation is this? It's a log equation You know how I know because there's x's inside the logs What don't I like about this? Because there is something I don't like about this. What do I have over here? Logs and You know first thing I would do I would rewrite this get all my logs to the same side and all my other stuff to the other side I'm going to plus this Over there and I'm going to minus this over there at the same time. I'm going to rewrite this as two log base six of x minus log base six of x plus three Equals one because doesn't that look a little bit nicer like what we often saw? What's my strategy now? Sorry, I think you're right, but I couldn't hear you. Did you say something about a two? Yeah, uh-huh Because I want to combine these but I can't right now So I would say move that there I'll have the log base six of x squared minus the log base six of x plus three Equals uno Now what you know what I'm going to need more room now what? Sorry Yeah, this is going to be the log base six of there'd be an x on top squared sorry an x plus three on the bottom It equals one Joel do the logs cancel Which one think carefully before you answer? No, if they did cancel what would I need over on the right hand side? And I don't oh, but where is the x sitting Joel? If I know one I know both What to the y equals oh, I think it's going to be Six to the one equals x squared over x plus three To me this is yeah okay now This is all great to me so far I have a feeling though that the quadratic that we're going to get Doesn't factor and requires the quadratic formula. That's overkill to me on your test though I think this would be a great question if I made this so that it factored And then you had to check for extraneous roots and either they both worked or one worked or none worked Right one date two dates, you know dates So the rest of this I'm actually going to pause and say What do you think qf stands for? Quadratic formula. I'm you know what I'm not testing that Okay, but you would cross multiply And you would get x squared equals six x Plus 18. I'm pretty sure if you bring it over you have a negative 18 numbers that multiply the negative 18 and add to negative. Yeah, nothing works. It's going to be yucky desmos. Where are we going? My gut right now is about another half hour ish, and then I'll do specific questions. Okay, is this helping? I hope I hope I hope okay. This is a nice way to do sort of the whole unit similar to the review quiz that we did $500 investment appreciates at 50 appreciate means increases At 15% random compounded quarterly That's a trigger phrase for me that tells me an awful lot Find the value of the investment after eight years, you know, I'm gonna do first What kind of a question is this compound interest? How did we handle these? Sorry, yeah, I'm gonna write this out because now I feel better. What are they asking me to find here? Right, what are they asking me to find here? More specific, what are they asking me to find here? You know what then this is going to be straight plug and chug no law because a is already on ground level Yeah, that's me to find final amount. That means I should be able to figure out everything else What's a zero here? Oh, okay. That's right. We'll do a little defect right a zero 500 do I have one of you for physics? No, I don't have you I Explained her what defect means on the day Defix stands for list your data find a formula insert the numbers crunch the answer It's a strategy for physics. What does dolp stand for Emily? Draw a little picture went in doubt defect and dope I stole that from a t-trip. No, I should mention that Um, hey, hey, hey, hey, what's C? It's going to be one plus Point one Five No, it's not Amanda over four Why over four? According Okay, so it's compound interest what I said and you guys kind of zoned out Appreciates means increases I said How do you know how do you know it's going to be a one minus decreases? Five fifteen percent would be whatever. Okay. I can't do that for compound interest though because compound interest doesn't decrease It always gets bigger Banks don't like to loan you and money and lose money Yeah, what do you mean by they you got to be very careful here The compound interest questions will always be one plus. Okay I can give you a population percentage question where it's one minus because the population can decrease Or I can give you a car value question where it's going to be one minus because cars go down in value sadly dramatically Is that a compound interest? okay Can I keep on? What's t eight What's p? I think the final amount Is going to equal 500 bracket one plus point one five over four To the power of eight over Point two five or eight over one quarter or dividing by one quarter is the same as multiplying by what? You know what eight times four that's actually going to work out to a 32 if you really yep. I don't know. Why did I? compound interest Okay, so I'll jog your memory. Are you ready? What's the interest rate for a whole year? What's the question say? What's the interest rate for a whole year? If it's 15 for a whole year and it's compounded quarterly, what's the interest rate for one quarter of a year? Divide by four. What if it had been compounded monthly? What if they'd compounded semi-annually divide by? What about weekly divide by 52? What about daily divide by 365? What I think that's all It's quarterly semi-annually monthly Sometimes daily sometimes weekly but not very often Okay, so it's divide to adjust the interest rate and the period is also a fraction It's going to be 500 bracket one plus point one five over four close bracket to the power of 32 eight over one quarter 1624 dollars and one That much How much was the interest? Well, if we started with 500 bucks and we ended up with that the interest was $1100 and 20 That minus 500 is the interest, right? Turn the page Okay So I said that we did we spent one day looking at logarithmic scales There was Richter scale Ph and decibels Decibels I like because I think most of you listen to music and listen to sound So nerdily I like it, but I'm not going to ask it on the test decibels was a bit tricky because it was decibels It was I get to divide by 10 Ph to me is sort of fair game. I'm not going to ask it on your test But ph and Richter scale were the same thing It was always going to be 10 to the power of the bigger number minus the smaller number Equals how many times more? What is 10 take away eight ph of 10 th of it, sorry You know what you with me am you are now Okay, and I think I've told you it's going to be Richter scale on your test 10 to the power of the bigger one minus the smaller one equals how many times stronger and I think I told you I'm almost certainly going to ask you to find one of those because that needs logs So I'll tell you how many times stronger the earthquake was Is that okay here 10 squared 200 in fact this I would argue could be on a non-calc section very easily I'm going to assume all of you can go 10 take away eight get it to and 10 squared. Yes For the purposes of what the questions that I'll give you because remember I went on a bit of a rant on some of the questions I said I thought they were poorly worded So I'm going to go with you know, what's bigger minus smaller is how many times stronger Is that what you're asking me Yo Because I'm stupid The caffeine's wearing off I think happy thoughts of faith and trust You can fly you can fly okay, but there's no place like home. There's no place, but I'm stuck here. Shut up. Okay Number 21 is a nice twist. It's an exponential growth But instead they're asking you to find don't write this down actually They're giving you everything but that Here's a base e question You know how I can tell at a glance that it's a base e question Okay, and I said to you if it's a base e question I'll either always give you the template or the whole equation I think in this case they gave you the whole equation. They gave you k is negative point zero two Oh, and they gave you hey, what's that? When what's that six nine six three zero initial amount, okay Um says in what year will its value drop below 3500? What are they asking you to find here? T What's this then? Now what did you say minus? No, you didn't because if you had I'll be freaking out right now I'll be lunging over the tablet computer at you right now with rage on my face You didn't say minus. Did you of course not? Uh, let's see 3500 divided by six nine three zero I'll probably go 50505 because I don't want to round off because when you round off with exponents Your round off errors increase exponentially in 50505. I can remember pretty easy So I get 0.50505 equals e to the negative point 0 to 82 t. How does that help now what? Uh ln both sides right I even left a little space in front so I could fit that in because I'm running out of room Uh I'll get ln of point five oh five oh five equals negative point zero two eight two t ln of e And the reason that's so nice it's still is the ln of e is one. So how would I get the t by itself? Is that minus I gotta divide you're right t equals ln of point five oh five oh five divided by negative point zero two eight two ln of that answer divided by negative point zero two eight two And I get 24.22 and I go What year did this start in? So what am I going to have to do with that number? Okay, okay fair enough plus 1998 during the year 2022 The value will drop below 3500 bucks I don't like that extra step because I'm too concerned some of them make a dumb mistake here So I'll probably start at year zero or just ask how many years until But otherwise that's a nice base e exponential growth question Is that okay? You guys doing good? Not getting spinny on me? Sure. Okay Number 23 What kind of a question? Well, this is I think an exponential growth. They change the letters a bit first of all. What's my base? two Where a is the final amount a zero's initial amount t is the time and oh instead of p. They're using d for the doubling period okay It doubles every 7.5 years. I think that's d How long I think they're asking me to find t What's the five? I'm afraid to leave here be with me. I'm tired too, but I'm sucking it up. What's the five? What's the final? Okay I'm going to not bother doing this because I'm sure there's going to be one in the written I would give this to you actually as a written question and I would say it doubles I wouldn't actually give you this I would expect you to know that look for me. I would say it doubles every 7.5 years If the starting amount was five dollars and the how long until it gets to be worth a thousand dollars So that's fair game, but I'm just not going to do it It can be written Oh, here's a Richter scale question okay Says the Richter scale is used blah blah blah blah blah blah blah blah An earthquake of a magnitude of five Is 50 times more intense than one of its aftershocks determine the magnitude of the aftershock Is this the strongest earthquake or is this the weaker earthquake reading this sentence carefully To the bigger minus smaller equals how many times stronger? That's the equation you're going to solve How will you solve it? Log both sides. Are you guys are you guys okay and solve this or do you want me to do this? So let me ask you the game because it wasn't a yes or no question. Are you guys okay in solving this? Or would you prefer me to do this? No, okay, thank you. That's good And number 25, I'll just get the equation for this is an exponential growth question. Okay Uh You know what? Forget that That's yucky Written what am I going to ask on your written? I've tried to give you some hints like there's going to be an exponential growth word problem on your written There's going to be a Richter scale on your written So in fact, there's going to be a couple of exponential growth word problems probably one increasing probably one decreasing Oh, no, no, no, no I said to you one a half life for sure and then one with a per population increasing or decreasing Okay, it's gonna be an if then on your written Okay, I think after that it's going to be an exponential question like this with x's as exponents Except not this tough because this has got a quadratic See how you're gonna get an x squared when I multiply these together and this actually is a two and a two So I'm not gonna let me just look if I got a ah So not like this Something like this Where you have to take the log of both sides Uh, how many logs do I how many x's do I have to get on the same side? It'd be wonderful. There's some kind of a grade nine mathematical operation where but okay could be one of those And then there's going to be two count them two log equations Something like that So let's do these last few here This is an exponential equation. What's my base here? What's my base here? Can I write a five as a two? That's why I got to use logs Comparing that with the one that I scribbled out or a base as a two and the 32 is also a two I could have done this one without using logarithms here We're going to use logarithms now. What makes this question a little bit tougher is there's a coefficient Easier is no coefficient So this will be about as tough as I'll throw at you This is where I said no shortcuts because easy to make a sloppy mistake And I have a feeling we're probably going to end up using two columns So I'm going to write over here first and then move over here because I'm going to do this in about eight lines What am I going to do first? What can't I do with this five and this six? Can't make it a 30 And unfortunately if you do that I I think I capped this This is probably going to be worth And I'll probably cap it at one Because that's just uh What will I do then? Where's the x sitting? So this question should be obvious. What will I do now then? Log both sides now going on from there is a tricky part, but for starters, I'm going to say, okay Log of two to the three n equals the log of five times six to the n plus four The most common mistake with these ladders people now move the exponent to the front on this side Why is that technically wrong? In fact, not technically completely absolutely positively wrong If you move this exponent to the front, what are you actually saying? Replies to the five and does it apply to the five? No, no What are we going to have to do then break this up? How? What's happening between the five and the bracket? My next line I'm just going to drop this guy down not taking any shortcuts Now what now move the exponents to the front What am I going to have to make sure of with this exponent by the way it's all brackets Guarantee you some kids won't on monday and they'll do the error that we saw in the one right. Oh Kills me Shannon question or it's a question to log Here and log six plus four you could I I am so worried about making dumb mistakes here I don't do shortcuts on these ones. That's all Go ahead, but you're just taking the gamble that you make a sloppy mistake, right? And you you have me last year in physics. Am I usually fond of doing stuff in my head and taking shortcuts? so I think am I not like in physics don't I do it in your head My my hint for you is then is hey do it because going all ballistic about not taking shortcuts Probably has a lot of experience with kids making sloppy mistakes on this and has a good reason to ready Three and log two Equals log five and yes now I would go and log six Plus four log six by going like that. How many ends do I have to? Get them the same side Easiest is going to be minus that over three and log two minus log six Equals log five plus four log six Double check to make sure I didn't make a sloppy mistake I forgot to am My subconscious was bugging me by the way guys like I stopped and said that because Something was bugging me My brain works weird Really? We hadn't noticed mr. Do it. How many ends do I have? How many would I prefer? It'll be wonderful if there was some kind of a grade nine mathematical operation that Okay, so I'm hoping me using those same phrases has given you a trigger so that when you ask the question I got two ends. What do I do? Oh, we want to I'm hoping that pops into your head So I'm going to get this and bracket three log two minus log six equals log five plus four log six Last sorry second last step. The last step is being able to calculate this Updating your status Okay, now what? Hey, all of you Dual included You want to practice trying to type that into your calculator to see if you can get the right answer? I'm going to freeze the screen so you guys can't see me Answer accurate to at least two decimal places Let me double check my math to see Log five plus four log six And it's typed in right Don't do that So hey folks You need to figure out what you're typing wrong then because silliness would be to do all the hard stuff And not be able to use your calculator properly and lose marks for that Found it. Okay, do it. I've typed this in three times and I've gotten three different answers. What do I do now? Move on to the next question, but You've finished the test You come back to this question You go y equals you clear whatever you have there You graph the left side two to the power of bracket three x As y one You graph the right side five times six to the power of bracket x plus four As y two you hit graph Uh-oh small problem What's my answer? You know what? I better change my x max to 40 Otherwise the answer won't appear on my screen if x is 30 Okay, I'll do that I'll go work on the next question while my calculator is thinking then Second function calculate intersection first curve second curve guess Am I right am I right am I right am I right? Seriously, I mean this is going to be worth four marks out of 40 For 42 ish That means that one question basically 10 percent of your test So what you're really saying is I just got perfect on one tenth of my test for sure That is worth growing your triumph a little bit Okay, sure. I'd rather have that Okay So I said there's going to be an exponent exponent equation like this And It's going to be a logarithmic equation. She's going to be two Oh, you know what? I'll even be more specific There's going to be one log equation like this with logs in everything And there's going to be one log equation where there isn't a log over here And you have to say how do I get the x by itself because the logs don't cancel? Oh, if I know want to know both Which we've already done a few of in in this test What's my strategy when I'm solving a logarithmic equation? I want to write it as one thing equals one thing double check though quickly alex or my bases all four Oh, then I can combine these The right side is already one thing. So I'll just drop it down The left side alex. What's adding two logs the same as So this is going to be the log base four of joel. Be ready Come on one's coming to be ready Joel Can I cancel out the logs here? Yes, why? Now technically I'm not canceling and taking the analog but shut up. Mr. Do it. Yeah, okay for all intents and purposes. We're saying I'm not going to do that because that's just a fence. It's technically wrong The next line though is this six minus x five minus two x Equals 60 Now what? Well, it's not factor factor is when you're bringing foil Front door bomber. You had Mr. Rocket clearly Uh 30 minus 2x minus 5x Plus 2x squared Sorry, what? Thank you gather like terms 2x squared minus 17x Plus 30 Equals 60 What kind of an equation is this? How do I know? Make it equal to zero i'll minus 60 from both sides In fact, I'll get 2x squared minus 17x minus 30 Equals zero There you go. I knew I could get you You're still not nowhere near my best Hailey fiddler two years ago was pretty good I had a student about five years ago named kevin Who was chronically sleeping because he was an esl student who was always Emailing us and talking to his friends all night So he like he fell asleep several times during tests and I had to shake him to wake him up He sat right where joel is right now And when I did my tarzan yell He would jump so badly twice out of the chair onto the floor He hated the transformations alarm bell He was not Oh, I don't like alarm bell. Mr. Dewick Well, that's a shame I don't like that. You're staying up all night now This factors these are the yucky ones to factor because of that How do you factor those you may have learned the box method? You may have learned factoring by grouping that's grouping joel not groping. Um, you may Do you guys know how to factor these? okay I'll give you a method because there's like five Okay Goes like this and you might want to jot this down and write down the quadratic somewhere. I think this factors for me I don't know. What did I say at the beginning of this tutorial? Okay, who knows how to factor What do you do? Can you tell me? okay That's enough for me to know which method I'll show so that also I'm not giving people who already know how to do something a new method that they might say Oh, I'll try that and then they get you know, okay Shannon here's what you do You find two numbers that multiply to negative 60 And add to negative 17 What are two numbers that multiply to negative 60 and add to negative 17? Well, let's see what multiplies to 60 first of all 10 and 6 I can't really get a 17 out of there. What else multiplies to 60? one and three Oh 20 and 20 Amanda 20 and three And what is 20 take away three? Can I get a 17 out of a 20 and three? Okay, so it's almost the same as when you had nothing here where you asked what multiplies to that and adds to that Yes, I'm here. It's what multiplies to those And adds to this. What are the numbers? So here's what you do you rewrite the 2x squared You rewrite the minus 30 But you take this negative 17 and you break it up into a negative 20x Plus a 3x That's what you do Okay Is that still technically a negative 17x? Okay Then what you do is you're going to do factoring by grouping not groping dual grouping Okay, what you're going to do is you're going to factor a gcf only out of the first two terms So temporarily ignore those GCf to 2x Watch and watch and left behind is going to be an x minus what? then Ignore these two factor in gcf out of those positive or negative Yeah, you're I just want you to say it Which reminds me to put a plus sign there. That's what positive three Full pull a three out of here. What do you get? Pull a three out of here. What do you get? You know how you've guessed right look at the two brackets. They are what? That's how you know you've guessed right. Okay, we're almost done. There is another gcf now There is something that appears in this and this the bracket What bracket appears in this term and in this term? What bracket appears in this term and in this term? It's now a gcf if I factor out an x minus 10 You know how I've left behind That's called the grouping method called grouping because you group them into tubes into pairs And I always do the groping jokes, you know, whatever stupid thing I can do to help you guys remember That's probably the second best method to do this Best method when I taught math 10 I used to give my kids a hundred of these And partway through their brain would hardwire a subconscious circuit and suddenly they would just start Seeing the answer. That's the best way, but it literally I would actually give them a hundred of them It was painful, but all of a sudden you'd say partway through Mr. Doock. I'm just looking at these and I'm seeing the answer. Yeah Your brain was doing it subconscious this whole process very quickly subconsciously our brain's a remarkable remarkable instrument You okay with that? What are the roots? That's only the phone call Hello, hello Oh, please. Hello. What's the answer? Put a 10 in there. What do you get? reject What about here? Well, what's a minus minus the same as I'm going to have six plus something. That's got to be positive What's a minus minus the same as I'm going to five plus something positive. I don't even care what the answer is. You know what? There is a date, but only one This is overkill That's two exponential growth problems and one that's saying how many years will it take until dbd equals vcr They want you to do two exponential growth and let them equal each other and forget it Are you guys okay on a half-life question? I can do this one if you want to because it's going to be on your test Okay, you got to go Okay, sorry no printout for you, but I'll email it out too as an attachment You okay, you okay Gesundheit gesundheit gesundheit Joel's going You ready Says it takes 15 years until five percent of the original amount of a radioactive substance remains Determine the half-life as soon as I see half-life I go a equals a zero c to the t over p and I know c is a half Because that's how half-life is defined. What do they want me to find here? The half-life yeah the half-life. Thank you captain obvious. Which of these is the half-life Okay, and this Shannon you got to memorize the half-life is the growth period the half-life is That that means they told me everything else. Um, oh What's this 15? What's this five careful? I'll give you a hint. It's got to be a final or an initial amount because that's all that's laugh Final amount as a decimal Is point zero five What percent did we start out with then if you're finished with five percent? What percent do you always start out with? What's another word for everything? What's the number a percent that describes everything? hundred percent Right, which as a decimal is what? one You're starting out with a hundred percent. You're ending up with five percent Is that okay? Let's go here. We go Point zero five equals one bracket point five to the 15 over p Is that one gonna make a difference but if there was a different initial I would divide I would not go times Where's the p sitting log both sides Uh, why am I going point zero five? No, yeah, it is point zero five good gosh point zero five Equals why am I going black all of a sudden mr. Dude? Point zero five equals and the exponents gonna come to the front and you're all gonna write a proper Big two level fractions so that you know what's on the top and what's on the bottom and you know what I'll even just this once do that Because Vlad, how do I solve this? Darn right. It's math eight Yes, it's ugly math eight still math eight You're gonna get this times this equals this times this or physics 12 students, which is everyone but amanda I'm gonna move stuff diagonally. That's gonna move to there on top. This is gonna move where Down there Do I care about the one? p is going to be 15 log point five divided by log point zero five it's gonna be 15 log point five divided by the log of point zero five The half life is uh, what does it say how many decimal places? At least two I'll go three point four seven Years Did I type that in right point five and point zero if I said it? Okay, you've seen 98 percent of the test I reserve the right for one multiple choice mini curve well right because again I'm getting used to university when you have to learn how to figure something out on your own Okay Let me right click are we done? I can stay longer really want me to okay I'm gonna right click and hit save and then I'll print up copies here