 Boy, I hope that gave it a different name Here we go The basics we started out first of all looking at exponents We defined the exponential function. I need that graph paper back the exponential function Was something that looked like this Well really that's what made it an exponential function the x was an exponent But we could add a vertical stretch to the front here just like a transformation stuff And we could add a vertical slide right here And we could add a horizontal reflection or a horizontal stretch or a horizontal slide right there Not going to throw all that at you. What we need to know is this now on your unit test We asked you to graph a couple of these by hand We won't be doing that on your practice Provincials around the real provincial what we will do is give you a question perhaps on the non calculator section and ask you about domain range x intercepts y intercepts things like that Every exponential equation looks like this if B was bigger than one if your base was bigger than one So this might be a graph of y equals 2 to the x or y equals 3 to the x or Y equals 10 to the x and we call these Exponential growth equations because as you move to the right it grows or They looked like this if B was a fraction for example, this could be y equals One third to the x or one quarter to the x or one tenth to the x No matter what they looked like they all had certain things in common Every as long as you didn't stretch or flip or slide as long as you had the basic exponential graph They all had certain things in common they all had a y intercept of Zero comma one and you can memorize that or you can derive it if I put a zero in for x. What's anything to the zero power? one So remove the graph paper because this is not to scale at all that point right there is zero comma one and that point Right there is zero comma one They all had a domain of All reels The x intercept maa No x intercept the range was y greater than but not touching zero Now you can decide whether you want to memorize this or if you memorize the graph I think those are pretty obvious from the graph and from the equation You can decide whether you want to memorize or derive it. I don't care But the reason I spent so much time on this was this allowed us to graph logarithmic functions logarithmic Spelled right. Yes functions the key was This was the inverse of an exponential It was the inverse of an exponential. How do I find an inverse? Switch the x and y's around in the equation or on the graph Big deal huge deal Watch watch watch watch watch remember all the stuff that I wrote for the exponential copy paste Now that's for an exponential. We're gonna switch the x's and y's around right oh Domain, uh-uh No, why intercept not range Domain x is greater than zero Haven't made a mistake yet The log graphs looked like this So if we had y equals log base b of x Where b was bigger than one by the way, do you notice in both of these Be as bigger than one be as smaller than one can be be equal to one No, because one does not follow our usual exponent rules. What's one to the fifth power? One what's one to the sixth power? One try really hard one. What's one to the twelfth power? Very good Absolutely boring doesn't follow our normal exponent rules. So we said our bases had to be positive had to be bigger than zero But not one that's the restriction because the other thing they will ask occasionally is Restrictions about log or exponential equations kids are pretty good at remembering most of them But the one now forget is the base both for a log and for an exponent can't be one because one doesn't follow our normal exponent rules What's the graph gonna look like? Well, let's turn the graph paper back on It would look like this. It would have an x-intercept of 1 comma 0 The domain is everything positive in other words the graph is going to exist only over here And if you take the inverse you'll find Log graphs look like that now you have to be careful those you that cheat and use your graphing calculators You're graphing calculators Because this approach is nearly vertical will often just draw the log graph like that and stop it in mid-air This keeps going down and down and down and down Now that's when B is bigger than one By the way Do you notice in this one when B was a fraction? That was the same as just flipping this horizontally Instead of getting bigger going this way getting smaller going this way, but it was a horizontal reflection What kind of a reflection? Horizontal what letters associated with horizontal by the way? x okay You know for the inverse for the inverse When B is a fraction it's going to be a vertical reflection the inverse of That graph there or in other words the log When base B is a fraction it's gonna look like that I'll write it in our notes, but it all ties together because a vertical reflection switching x and y around for the inverse even the reflections which is around Graph paper So if I have y equals log base B of x where B is As a fraction between zero and one it's still gonna go through one comma zero It's still gonna have a domain x greater than zero. It's gonna look Like that so what are we gonna ask you about this We're gonna ask you to find things like asymptotes domain range and intercepts and Let's be pretty clear I'm probably not going to ask you for the domain of this one here aureals Probably not going to ask you for the range of these two here aureals This one I forgot to mention also has an asymptote The asymptote is y equals zero. It's a horizontal line zero high and You know what if it's an inverse Logarithm will also have an asymptote x equals zero a vertical line My brain gets tired of memorizing stuff So I've always just memorized the exponential inside and out and then since I know that the log is the inverse I can almost always derive whatever the heck they want me to find by being reasonably clever We'll do some specific questions in a bit Then we defined the log operation Mathematically log Definition you know what that's wrong. Mr. Duk. You should write it like this log Definition when you email someone and it's all caps. What does that mean? When you email someone and you write in all caps in internet in the netiquette as they call it. What does that mean? It means something very specific means you're shouting This is the key to the whole unit This is also a question. I guarantee will be on Probably every single one of their mock exams and it's one of the few questions that I can guarantee you'll see on the provincial The log Definition, so I'm putting this in caps because it's usually important. Here's what it said If you knew that a equals B to the C You also knew that the log base what of what equals what the log base be of I've heard both C and a which one And I'll be brutally honest if you don't have that at your fingertips by the time the mock rolls around You'll probably score on the mock about 20% lower than your current letter grade Having just done this before I know what the key signs for someone who's unprepared are this would be one Not knowing that sign is y over r cos is x over r tan is why that's that's one of my other little trigger signs I know in each unit what my triggers are when I talk to a kid because I'll be doing tutorials Right up to the mock exams and when I talk to a kid and I can't remember that I go They're gonna drop 30% on the exam If you get the basics and then some then what you're doing on that first mock is just getting your dumb mistakes out of your system And you're all gonna make plenty trust me These are interchangeable if I know that I know that this is the logarithmic version This is the exponential version So as a great question as a great multiple choice question We'll give you something. We'll say write the following as a log or write the following as an exponent We did a lot of work with numbers for example, we would say something like this the log base 4 of 16 What's that? Well, oh to By the way, do you see that if you know that you can derive this thing in a pinch? Because can you write the exponential version of that what to what power equals what? 4 squared equals 16. Mr. Kamosy is quite right I picked a bad example because 2 to the 4th also happens equals 16 So you know what instead of using 4 and 16. How about I use? 381 and 4 the log base 3 of 81 is 4 now 3 to the 4th equals 81 But 4 to the 3rd equals 81. I actually made up one that you could do wrong and technically you got a right answer over Here we'll do that one again. Yes, I will I do it every year Okay, then we came to what we called log rules the log rules What I'll do is I'll do an example and then I'll do the generic one beside it So you can kind of see a before and an after so one of the things we said was something like this the log base 6 of 18 Plus the log base 6 of 2 so it would say Simplify are my bases the same? Check because if they weren't I can't use my log rules The only one I could try pulling out is what we call the base change law, which I'll do a bit My bases are the same. Okay, what was adding two logs the same as Yeah, now if you're not sure first of all, I'm hoping someone said multiplying I gotta repeat that for the benefit of the microphone here. Don't I but multiplying if you're not sure Remember your grade 9 exponent rules when you were multiplying. What did you with the exponents? Add and if logs are the inverse of exponents. I guess that means if you're adding what do you do? Multiply And multiplying this is the same as the log base 6 of 18 times 2 Can someone please in their head? God, please do that. What is the log base 6 of 36? Generically we said this The log base B of x plus the log base B of y Equals the log base B of x y Okay, could you get that door for me actually? Just shut it all away. Thank you again Log rhythms, I know the unit is tough, but at the same time it's very symmetrical Hey, if adding is multiplying so if adding outside the log is multiplying inside the log What do you think subtracting is? Dividing if I have something like this the log base 5 of 50 minus the log base 5 of 2 That's the same as the log base 5 of 50 divided by 2 Yellow and what is the log base 5 of 25? to Mr.. Dewick, I think every single answer just works out to 2 no It's because I'm just too tired to make up tough yucky questions Please don't look for that kind of a pattern as a log rule. We said this The log base B of x minus the log base B of y Equals the log base B of x over y and how about I call it example 2 like it should be Example 3 for some reason this seems to be everyone's favorite log rule. It's the one they remember the most Except I'm gonna ask it backwards 5 log base 3 of 7 equals And what I'm really asking is What could you do with coefficients? Make them into exponents and vice-versa sometimes we wanted it to look like this that was better Especially when there was an x up there We wanted to get the x down to ground level remember doing that taking the log of both sides for exponential equations, which I'll get to Sometimes to use our log rules We need to move the exponent back into the log and don't worry. I have no idea what 7 to the 5th power is But it does give us our log rule our log rule was I think we said a log base B of x is the same as log base B of x to the a we called that the power rule So we would give stuff like this then simplify the following one half log base a of B minus log base a of B squared plus log base a of C minus log base a of Oh heck minus two log base a of a cubed This is my worst unit for making up questions if I'm during the unit. I'm in a zone, but right now What the heck did we think? simplify are my bases all the same check all base a's If I want to use my log rules can't have that guy Can't have that guy got to move them up B to the one half minus log base a B squared plus log base a of C Minus log base a now it's going to be a cubed and then I'm going to move the squared onto there What was my power to a power rule? Multiplication just to save room then I'm going to eight of the sticks now the rule of thumb I gave my kids was once you've got rid of the coefficients Multiplying is the same inside the log is the same as doing what outside the log adding and Dividing inside the log is the same as what? Outside the log subtracting so here's what I just tell the kids if it's positive put it on top if it's negative put it on That in other words really quickly now. I would say okay. I know it's log base a positive on top negative On the bottom, but now I've taken care of the negative positive on top negative on the bottom By the way, it may be in this question that they want me to simplify this because I could also Move this six to the front and what is the log base a of a everybody? One that's just a six But that would no longer be inside of the log if I gave you this question It'll be multiple choice and you would look at your answer as you would say Oh, I don't see an a anywhere in my answer, but they got a little minus six over there Okay, they probably took that a out of the log I I don't know it's kind of hit and miss depending on how far they want you to simplify I do know they wouldn't leave this like that because I have bees on the top and I also have bees where on the bottom As soon as I said bees where a column look to original No letter So how many bees do I have on the top grand total half of one? How many bees do I have on the bottom grand total to? How many left and where will they be? I think I'm gonna be left with one and a half on the bottom Right a half on top two on the bottom. I always just think in terms of cancelling for exponents So I'll have a C on top I'll have a three over two or a one point five. I'd look at my multiple choice answers. How do they want me to write it? I don't know a to the sixth. I like this question. I like this question. I like this question Why would I say that you didn't hear me say would be on the test You're just good students One of these is easy one of these drives me crazy because it's easy, but kids don't find it So here's the easy one. I give them so we're still under the simplify instructions and we give you something like this Log base a of a To the 12th this one kids have no trouble with they say oh The 12 moves to the front, and I have log base a Of a and what is the log base a of a one? And they say that's just a 12 I'm gonna do that same question, but I'm going to do use different words This is an easy one. I'm taking the log base a of 12 and raising 12 as an exponent to the power of a Sorry, I'm raising a to the 12th power Exponents and logs are inverses those two operations if they're inverses should just cancel each other out and the 12 should Be left behind now. Why did I say it that way? Here's the one that's equally as easy, and I try to be crazy The same question No, they're not. Yes, they are in this question Am I taking the log base a of 12 as well as raising that 12 to the power of a Yes, in this question Am I taking the log base a of that 12 as well as raising 12 to the power of a Same question now One of these is easy One of these I like way more Capiche First test dude Good not gonna be that easy probably I can never resist like I'll throw something scary in there instead of a base a how about a Base pie like that really terrifies this not so still 12 if your exponent and your base are the same What drops down is just that you can prove it But this is one of the few times I say memorize that one this one I think almost all of you have just because of the power rule But it's the same thing as this Isn't that cool example six here was our next rule the base change law This allowed us to change the base of a logarithm And I could either give you the front end of the law and then ask you to give me the back end Or I could give you the back half of the equation and say where'd it come from There was that second variation that people had more trouble with show you what I mean the log base nine of No, that's gonna have another answer as a two again. They're gonna go crazy. I need to have a different answer. Mr. Do it No, 125 all over the log base nine of a nickel five How many of you glance at that and know right now the answer is three one Two one liar Here's what I want. You know spot the pattern. What does this fraction have that makes it interesting? What's the base on the top? nine, what's the base on the bottom? This gave us our algebraic base change law. We said look if you have a fraction and I don't care. What's there and I don't care. What's there, but what do they have to have in common? Same base That's the same as Now I'll do the numbers over here. This is the same as the log base five of 125 that's the base change law By the way, five to what power equals 125. What is the log of? Base five of 125 five to what equals three Three was sitting in there in the first fraction, too If you were good at your logs Why can I rewrite it? My original bases were identical. I can write it as any base of I wanted to sorry log base y of x Now where did we use this? Here's another question. I like this question and you do need to get your calculators out for this one if you're following along at home the log base seven of 800 If I ask you to evaluate that What did you type into your calculator? You remember log 800 over log set what you're really doing is the base change rule Let me show you here's what you said This is the log 800 over The log seven if I don't write a base which base is there automatically we called it the common base look look look You have a 10 there and a 10 there You see how that and that that that's the same thing It's just you're way more comfortable with that one because you've typed it into your calculator so often By the way, if you haven't noticed by now I'm trying very hard to tie stuff together in logs because I think the big difficulty at the end of the log unit is too Many kids have tried to memorize Everything as a separate entity. I'm trying to say not it all ties together. I got your email by the way Welcome back Where were you so was miss Lee's? I'm gonna get madam Galvin. Sorry. I got to get the last name. She said it snowed What was our weather like Saturday and Sunday boys and girls? How many of you are wearing shorts on Saturday and Sunday all of you really? Snow oh Boy by the way, what is the answer to it? Let's say three decimal places 3.435 I'll even bet you because your mock exams are going to have a 16 section 16 question non-calculator section probably the first calculator question. It's often this one By the way, let's talk exam strategy when you're writing your mock provincial you do the non-calculator section first And double-check your answer because you're not going to get that one back. You're going to turn that in But if you finish it you don't sit there Go do the trig identity. Did you do the calculator for the trig identity? No, it'll be last question go do the trig identity and do the transformations questions Did we ever use a calculator for the transformations question? no Then start working your way through the multiple choice and I bet you'll find about half of the remaining questions You still don't need a calculator for anyways In other words, what you don't do is just sit there and you know Okay So this is Is the base change rule in all its glory? This one kid seemed to get and I really got to be honest I don't understand why they have such a hard time extrapolating it to that Maybe if I wrote the 10 there in there That might make it easier to see but I'm not gonna Because it's a common base I figure going through the whole unit review is going to take me till just around I'm guessing four o'clock and trying to budget then I'll press pause and I'm probably gonna have to start a second recording And that second recording will go through the test that you guys have in front of you I'm willing to stay till five ish or so if I have to if you can't stay that long I guess it'll be a two-parter online on that bidler website common mistakes This one let's use numbers. Mr. Duke Now what's the? multiplication rule Multiplying inside the log is the same as adding outside the log Doesn't work the other way around or I'll see it this way instead of a plus sign there See a time sign that not no no no no no in fact think about your bed mass What would your first thing that you would do here actually be from grade eight? What should that be? More specific please what should that be a 22? Which is nothing to do with this so that's common mistake number one Should have left that whole thing there mr. Duke. That was kind of silly Also the same for the dividing rule so the number of times I've seen this log base 10 or sorry log base x of 10 all over log base x of Five That does not I'll put it does not equal up here to Put the plus sign back that does not equal by the way are my bases the same in this fraction Oh, that is a base change form What it does equal So that's yucky What it does equal is the log base five of ten, which also is not two by the way What is it? I don't know go log ten divided by log five whatever the heck that is but I need to It's gonna be one point three or something So that's common mistake number two. I'm pausing because there was something important I wanted to remember before I moved on to exponential and word problems, and I kept base e. Thank you brain ha Face e. I think that I shall never see a number quite as nice as e What was he Like the fifth letter of the up no in math stands for Euler's number or Euler's number if you're an American and refused to pronounce things like everybody else True down the south they'll call it Euler, but talk to the German students. It's Euler. I know that's how they would I'm serious But it's not the hockey team Euler's they spell it differently Base e was well, you could actually find it on your calculator, which I should really bring up I suppose You could find it on your calculator Right there second function e enter e is two point seven one eight two eight one eight two eight It looks like it repeats. It doesn't just off the page. It goes all haywire e belongs to the same category of numbers as pi It's irrational. It goes on forever and ever without repeating Now there are lots of irrational numbers, but like pi it seems to be a number that the universe really likes It's a number that describes it seems how Life reproduces or changes or grows or decays Do you remember how we wanted to show or what we used to show that we were dealing with base e? What kind of logarithm? How did we write it? Yes, L n I Always hand write the L. Otherwise. I'm worried that if I just draw a vertical line. I'll think it's a one I'm stupid that way so L n of 17 for example, you would go L n 17 2.833 Dot dot dot actually goes on forever Why L n? Well, it's actually French. It's log that's it out, which L n instead of nl That's fine. It's a natural logarithm now. What do you need to know about base e? Well that that I just did needs to be able to find the L n of a number You're going to need to use base e to solve an exponential growth equation, but I'm going to do that later on There's the log rules. Have I missed any that you can come up with in your head? I Haven't talked about if then questions, but I figured those would come up in the practice test that we go through Then let's move on to exponential equations Exponential equations were equations where the x was an exponent two main types The ones that I can ask you to do by hand with no calculator hint hint nudge nudge And then the ones that you need to use a calculator for and before I look at exponential equations I have to look at one little subtype what we called rational exponents If we gave you something like this x plus 5 To the 3 over 2 equals 8 This isn't a true exponential equation because an exponential equation I think by definition the variable was an exponent, but we did it at the same time What's yucky about this what bugs you about this what don't we like about this? The exponent we had a way to make the exponent Vanish, and it was very clever See remember what we did Both sides to the power of the reciprocal. What's the reciprocal of 3 over 2? Now why was that so nice? Well on this side? I have my grade 9 power to a power exponent rule What do you do with those two exponents? Multiply them and you know what three times two is on the top six Do you know what three times two is on the bottom? You know what six over six is? Do I even bother writing one when it's an exponent in fact in one fell swoop I have that Now the right-hand side Just in case this was on the non calculator section You don't what block are you done? What block are you? Can you put it block after me, please? Thank you, sir I'm not going to get to it tonight. I'm reffing for three hours, but I hopefully will get to it Maybe tomorrow Good luck. Thank you, sir Fraction exponents had an awful lot to do with roots eight to the two-thirds is The cube root of eight Square I gave my kids a stupid little mantra or rhyme to remember I taught my students flower power Remember that stupid little thing it said this if you look at eight to the two over three and let my grade 10 teacher Showed me this and I'll steal it She said Calvin that looks like a flower There's the flower. There's the root of the flower flower to the power root to the root Flower power Which because everyone knows this to go somewhere and this three goes somewhere the only question is where did it go? You had a fifty-fifty chance of guessing right, but flower power helps me keep it straight Hey, by the way, what is the cube root of eight to you say very good? Squared At least you got that I get x plus five equals four What does x have to be please God in your head? negative one So these were rational exponents questions Both sides to the reciprocal exponent both sides because your equation solving rules at God is ending both sides Then we did exponential equations and there was two types There was questions that we could rewrite by changing them all into a common base And then there was ones where we had to pull out the logarithm Here's the first type eight to the three x plus one equals 16 to the four minus x What's the base on the left-hand side? Hey, very good What's the base on the right-hand side? Okay No Actually, I would disagree with you I think the base of both of them is actually a two in disguise Because I think I can write eight as two to some power and I can write 16 as two to some power and Then once we have our bases the same the magic phrase was we equated the exponents We said oh, then the bases are the same the exponents gotta be the same By the way, you are right the base is eight and the base is 16 But I think right away I look at that and I see two screaming out of you Okay This is really Two to the third Equals two to the fourth What's my power to a power rule? multiply multiply In fact, I'm going to get this two to the Nine x plus three Equals two to the 16 minus 4x Yes, are my bases the same? What's my base? What's my base on the left-hand side? I can't say names and I feel bad that I said yours because I'm posting this online And there's a teacher technically we're not supposed to put any names online for some obscure legal reasons, so Sorry, but I figure yeah your folks like me so whatever What's the base on the left-hand side now? What's the base on the right-hand side now? Are my bases the same? Then you know what I can say about these exponents. They have to be the thing Mathematically what I'm really saying is my equation is this right We turned it into a grade nine Are you okay on solving that I think you're gonna plus 4x to both sides might well, okay? Let's do it real quick plus 4x to both sides. We'll get holy smokes is gonna work out evenly Wow Thank you very much. You've been a good audience try the deal That was a total fluke though I Should I should have just totally pretended like I'd done that on purpose they would have been pretty impressed Well, some of them would have been and I think you'll minus I think you're gonna get 13x equals 13. Don't you? minus three from both sides X equals one this would probably appear as the multiple choice Non-calculator section question. What's going to appear as your written section? Well in the log unit there are three types of written questions and since you're writing three or four mock exams Guess what you'll see each one of these on one of your exams the first type is Solving an exponential where the bases cannot be rewritten Identically something like this example This would be a fairly simple one. I'm gonna gradually build up an equation So don't write this first one down if those of you that are writing this down instead when I say, okay, write this down Here would be a fairly simple one five to the X equals 10 Can I write this as a tent? No a few times but no no not for exponents. I Could take the log of both sides Too easy, I'm not gonna give you this as a written I am gonna really really tempt you though I'll put something like maybe a six here and maybe a 12 here because a whole bunch of students will want to go Divide by six divide by six and make that a two, but I'm probably gonna give you a little bit of work there and maybe a There and if I really want to be really nasty, I might put a coefficient right there Except you know what I would really do to really tempt kids What would I put here? Mr. Kamosy to really tempt them to make a sloppy mistake not a five I Would put a two right there to really tempt them to turn that into what? When you can't Why can't you remember your bed mass? What does the east stand for? Exponents I'd have to deal with that exponent before I could somehow combine those I think here is the question that we're going to do together. I'm just gonna put a simple x there This is the one we're gonna do. Can I write this as 12 to something equals 12 to something? No, you know what I need to do because I'm so handicapped those you that have me as a teacher You know, I love to use first names I'm gonna call every one of you Johnny and just point that'll work because now I can go back to regular teaching style So Johnny, can I write this as a 12? No, honestly was throw me off my game So what am I gonna do first of all? Where is the X Johnny? Okay, if the X is an exponent and I can't write common bases what log of Both sides the first and folks those of you that have me know I like shortcuts don't take any shortcuts in this Almost every one of you gets the basic concept Almost every one of the shortcuts makes stupid mistakes, and I'll call them that that's what they are So I'm not going to take any shortcuts. I'm actually going to rewrite and in fact I think you got a half mark for doing that Usually these are out of four or five Little tired there Johnny, sorry Little low Little just clipped. Yeah. Oh That was way high. I Betcha it's frying the microphone Pardon me it doesn't do much. I was aiming great earlier today. There we go. Okay. Now, let's get There we go Oh, hi the low Okay, there's only been a few of you that haven't shot yet, right? Okay, the rest of you have me Right there, and I gotta shoot Johnny back there. There we go. Oh, I haven't shot you yet Well, hi There we go. Hey, you're all awake now. Everybody else have been shot by me at least once this year probably now what? Right-hand side is easy because our whole problem here Johnny is You know what the X is up as an exponent. We have no mathematical operation for solving for it Oh, we do we designed the law to be the inverse of an exponent. It can move it down to ground level Now a lot of students want to move this to the front right away. Thank you Can you put it on top of the appropriate block? Why would it be wrong to move this to the front right away? You would be saying that not only was this exponent on the six you would also be saying this exponent was on the To is it? No, oh What's happening between the two and the six mathematically if I was to write a mathematical operation right there Would I write adding subtracting multiplying or dividing? Multiplying what's multiplying inside a log the same as outside of a log? Adding oh, this is all coming together now here by the way. Here's the beauty if you get this Those you that did get this it's almost five free marks on your test It's gonna be a five marker or a four marker on your provincial on your mock on the real thing If you know how to do this it's basically be careful. Don't make sloppy mistakes, and it's almost free marks Now what oh? Let's move this guy Down now we can except I'm gonna have to be a little bit careful a lot of you just do this What what what? Now a lot of you don't write the brackets But they're in your mind you've got to be careful at those you that are writing the provincial You may have a marker who deducts a half mark for that because that's wrong That's what you've just written here is totally incorrect even if you bring them in on the next line This is an incorrect line brackets Now what what did I just write in here got to get rid of the brackets darn right? I'm gonna go like that and like that, so I'm gonna get this log 2 plus 2x log 6 plus 5 log 6 equals x log 12 Most people that get to this line get full marks Oh, sorry. I just lied unless they don't know how to type this into their calculator at the very very end I'll also see a lot of four and a half out of fives But you've done the hard stuff pause You see if you go carefully Carefully all you need to remember is your log rules and how to start take the log above both sides Now what? Get all the x's to one side get everything without an x to the other side I have x's on this side. I'll move this guy over here. What's sitting in front of my ex Johnny? What's sitting right right there in front of the 2x? So how am I gonna move it over? Minus, okay good old grade 8 math, right? Trust me. It helps my rhythm. I'm not joking. So I'll have still a plus 5 log 6, but I'm gonna have an x log 12 and I guess I'll minus 2x log 6 UK with how I got there and we come to the single key step your two lines from being done Who remembers well first of all what don't I like about this? How many x's do I have on this side to? How many would I prefer one? How? Factor the math f word not the f word you're thinking of factor Thank you for coming Because why is that so crucial? Why is that so nice? What's happening between the x and the bracket here Johnny? What's happening between the x and the bracket here? Mathematically multi say multi boy Sorry get the hand away from your mouth. I'm dead here multiplying. So if I want to move this over What's the opposite of multiplying? Oh? Perfect. I'm gonna get this Equals I'm gonna have that log to This someone else just leave you okay. Don't tell me who because I can't put it on the video here all over You see how that all came together You okay on this line here Well, here John Here what's happening between the x and the bracket Johnny mathematically? So how would I move this whole thing over? What's the opposite of multiplying? You said it say it again divide And then I said I like my accent this side right just a personal preference And the last thing all of you Every one of you no exceptions right now get your calculator out because it ticks me off How many kids get to this line? And then can't get the answer Johnny over there are possibly being one of them So how would you type this well? First of all since I have a top and the bottom I'm going to put the entire top in brackets and then divided by and the entire bottom in brackets The first thing I'm typing is bracket log to Oh, I noticed it opened a bracket in front of the two. I better close that off Plus five log six I need to close the bracket off for the log Did I just finish the entire top? Okay, sorry. Did I just come to the end of the top? Yes, so now I need to close that bracket off Divided by open bracket log 12 close bracket minus two log six close bracket for the log close bracket for the bottom fraction and That there is the correct answer negative eight point seven nine I guess if we round it off to two decimal places properly You know what that's important enough that I think I'm going to actually plunk that right in here Come on Clip there now you can even see what we typed in isn't that cool? That's an exponential equation That's written type question number one and it shows up. I Would say about 30% of the time on the provincial What shows up another 30% of the time maybe even 40% of the time is Word problems which involve solving an exponential equation in other words if you can't do this You can't do the word problems get and then the other thing that shows up not as often is a logarithmic equation We're going to do one of those next Only this time we're finally going to crack open This thing can you turn in your little test booklet, please to page to page? 30 40 30 Because I can't make up log equations very good very well. I know turn to pay 41 number 28 page 41 Number 28 how can I tell just by glancing at that that's a logarithmic equation? Because what appears in it? Are My base is the same My goal for any logarithmic equation is I want to write it as one thing equals one thing One thing equals one thing and I want to do that by using my log rules What's adding outside the log the same as I? I can write this as one term log base four of Six minus X multiplied by five minus two X That equals the log base four of sixty Mr.. Do it shouldn't that minus put that X on the bottom No, no, no, no, no, no remember my common mistake is people want to minusing or plusing inside a log They want to turn that into multiplying and dividing no no no no But here's what I do have Johnny. Do I have one log equals one log? Then you know what happens to those logs? I'm gonna say they cancel I'm being sloppy technically you're taking the anti log of both sad okay fine they cancel My new equation they've heard that line a gazillion times my new equation is this six minus X five minus two X Equals 60 now what foil shirt I'll get 30 and I think I'll get a minus 12 X and a minus 5 X minus 12 I think I'm gonna get a minus 17 X and Plus 2 X squared equals 60 what kind of an equation is this It's a quadratic. How do I know? Got it well no not exponents specifically a quadratic it's got a squared okay How do I solve a quadratic first thing before I do anything else? Gotta make it equal to zero It's gonna be 2 X squared. It's gonna be minus 17 X when I minus 60 from this 30. It's gonna be minus 30 Equals zero And now you have to be able to solve a quadratic equation. I Don't care how you do it Johnny if you want a quadratic formula knock yourself out If you want to cheat and use the quadratic solver I Have no problem with that. I've given most of you guys a quadratic solver However to hedge your bets to make sure you don't have an old-school marker just quickly jot down And if you write your roots on the next line, they got to take it I'm telling you every year that I've marked a question like this There's always been an old-school 60-year-old Marker who said how can we not give marks for solving this the kids can just solve this on their calculator? Every year they've lost that argument Just in case they don't lose that argument if you write that down They have to assume you were now good enough to just type that into your calculator and get the roots. I could I Mean B is obviously seven negative 17. There's a there's I could go type it straight in By the way that quadratic formula it is on your formula sheet somewhere, so you don't even have to have it memorized I'm going to choose to factor Because I can I'm looking for let's see does it factor numbers multiply to negative 16 at the negative 17 Yeah, negative 12 and negative 5 it's going to factor into 2x x I'm going to get a negative 17 out of there. How am I going to get that? Mr. Duke Five and six. Yeah, there we go. No, that's wrong. Mr. Duke. It's got to be 20 and three. Mr. Duke, how's that? I think that works if you foil it out By the way, how do I factor? I'll be honest I've done so many of these out at my mind go blank and somehow it guesses and checks and arrives at the right answer I'm not joking when I used to teach math 10 honors remember math 10 you learn how to factor these What I used to do with my honors classes I would give them it was bad, but I would give them 100 of these to factor By about the 50th one somehow their brain had rewired itself where they were factoring them in their heads and your brains Just work that way Well most years My roots are negative 3 over 2 and positive 10 Negative 3 over 2 am I right by the way Like you get negative 1.5 and positive 10 when you cheat. I mean use this use the technology I'm good with it. I by the way, I've been the teacher at that table each time that said look Solving a quadratic is math 10. Why are we testing them that? Do math 12? So but I'm not marking this year. So who knows It's an argument that happens every time Johnny If you stop right now, you just lost a half mark Why? You have to check for extraneous roots Right, you see if I go back to my original equation this 10 right here What's six take away 10? It's not for Six take away 10. Sorry what negative 4. Can I take the log of a negative number? No, no We reject Much like when Johnny goes out on a date. It's been a basketball player for years Johnny. So sorry You're the only one in the room right now okay This one I think if you go 6 minus minus you get a positive and I think if you go two times I think this one also ends up being positive if you plug in a negative 1.5 right there This one is a valid solution, but you've got to check for extraneous roots It by the way, I'm kind of re mentally vamping here. I'll get through an entire review I don't know that we'll actually get to going through that practice test. I May if I have no life sometime try and sit down on a weekend and just do a screencast of me going through that practice test No, because I'm also working on my master's degree and other stuff I think this is still being helpful for you anyways By the way, also get at your graphing calculators on this previous one on the exponential one on example two You guys remember how you can check? And I have no problem if you check your answer. This will be a calculator based question You would graph as y1 The left-hand side Graph as y2 the right-hand side and where these are equal is where those two graphs do what? cross clear clear clear Y1 I would go 2 times 6 to the power of bracket to x plus 5 close bracket Y2 I would go 12 to the power of x entered and then I would go Zoom standard is usually a good window There's the first one There's the second one. I don't really care that I can't see where they cross I just need to know can my calculator find intersect remember second function count intersection first curve Just hit enter a second curve just hit enter then it wants you to make a guess. What do we get for an answer? To example to what do we get? Make negative eight points that you know what I'll guess negative seven just to get it close It's gonna think about things for a bit Tells me the answer is that is that what I got when I did it by hand as well Well, great. So now I know for sure that I'm right That's a nice feeling on a test Especially on the real provincial Five out of five. That's five marks out of 90. I'm guaranteed on woohoo. It's a great feeling word problems word problems hoping desperately that I can get this file under 500 megabytes Otherwise, I'll be able to upload the screencast Johnny we're okay Yep, so we had Exponential equations logarithmic equations Word problems Take a look at number 29 on the next page Okay, the number of VCRs sold last year Was one thousand 120,000 and is decreasing at a rate of 18 percent per year The number of DVD recorders sold with 7500 is decreasing at increasing at a rate of six percent per year If this pattern continues how many years will it take I do like this question to do But because it's got to if this is actually two word problems in one I just noticed that to save time. I'm gonna find a different one. Just give me a second here. Let's look at number 21 I'll go back a bit got back a couple questions 21. Sure. This works No, don't like it shoot 22 is base e which I'd love to do a review of but I'm not sure I'm gonna get that in oh heck Mr.. Do it just make one up. Okay word problems The value of a car decreases by 23% per year if the car was purchased for $43,000 clearly not by a teacher How much is the car worth after 42 months Does anybody remember the magic exponential growth equation? I know I made my students memorize it. I'll give you a hint final amount equals Arrow that would use yours was I think ours was C to the T over P or something like that. I think that's the way the textbook said it Do I teach you? Do I teach you? Oh? You're using the base e1 Sorry, teach too many kids and I'm tired Johnny. Okay You're using this one for base e To the RT or something like that And you're getting it mixed up with the arc length formula. That's the next review Need to know this a equals a zero C to the T over P where a is your final amount a zero is your initial amount This is your rate This is the time. This is the growth period Pardon me It's not on your provincial exam formula sheet and it's well worth memorizing. Yes What are they asking me to find in this question final amount? That must mean they gave me everything else. Did they give me my initial amount? What's it worth? $43,000 They must have given me my growth or decay rate now decreasing by 23% That means my rate is going to be 100% Minus 23 minus because I'm decreasing. I'm losing money 23% we always start out with per percentages. We always start with a hundred. What is it? 0.7 7 as a decimal Time I got to go in years. So 42 months. I think 3.5 years. It's three years six months 3.5 years Right 36 months would be three years and then I added six more. That's why I was thinking about it man Now the period is how often per year this occurs Oh once per year. So the period is going to be irrelevant. It's going to look like this This is going to be straight plug-in chug a equals 43,000 0.77 to the power of 3.5 over one forget it. Do I have the a by itself already? This is straight plug-in chug clear 43,000 Times 0.77 to the power of 3.5. I get 17,226 dollars and 6 cents 17 226 point oh 8th mr. Do it oh six That's not what you're going to see on the written multiple choice. Maybe but that was to jog your memory For this guy. Here's something more realistically what you're going to see on the written section example to a substance Decays from 154 grams to 31 grams in 82 days What is the half life of the? substance And I think this would be the last one that will do and we'll wrap it up I didn't quite get the base e but read your notes. I Did a little bit of basic What's this question want me to find? What is the what? Half-life first thing I would do right away is write down if you're done. Thank you for that there a equals a Zero c to the t over p now Am I finding time or period? Because half-life is the growth period But by the way t is total time 82 days is the total time that means they gave me everything else What's the final amount did they give me that? 31 What's the initial amount? 154 What's the total time? Now in the previous question, I gave you a percentage and they may do that But this is a half-life question. You know what that means They're talking half-life what they're talking about as a growth rate of point five and they want me to find that It's going to be 31 Equals 154.5 To the 82 Over p Johnny where is the variable sitting? in and In and it's it's bonnet. You know what take the log to both side, but not right away You know the first thing I would do before I took the log of both sides Move that 154 over how? divide it 31 over 154 does that work out evenly? Can somebody put in their calculator? How about point two oh three? Sorry point two oh one three. That's pretty good So we have point two oh one three equals point five to the 82 Johnny, where is my variable sitting? No, where is my variable sitting? Yeah, I don't care where in the it's an exponent log log Now when I do that This guy can move down to the front. It's gonna look like this The log of point two zero one three equals It's gonna be a log of point five And here's where a whole bunch of people make life way too hard. Don't write this next bit down They write this So unnecessary I had a great eight teacher who was very good. She drummed into us all fractions have two levels write that down Because now Johnny where is the variable in the bottom? In fact Johnny, I can do this just extend the bar By the way What's down there? Invisibly I can cross multiply the salt. This is great eight now I think p is gonna end up being 82 times the log of point five divided by the log of point two zero one three I did the cross multiplying step all in my head to repress for time Sorry, it's gonna be 82 log of point five divided by log of point two zero one three closed bracket You know what the half-life is 34.46 days Every 34.46 days this decays by half loses half of its 35 that's what I said. Can't you listen Johnny and disappointed in your listening skills? Yeah What did I say it was thirty five point four six? Hope that kind of come back Okay, I'm gonna press pause