So, I started last day by showing you the intersection method because when I see this, what I really see is where does this graph hit this graph? And I know what this looks like. This is the line y equals 12. This is a horizontal line, 12 high. If I were to graph those on my graphing calculator, graphing that as y1 and that as y2, what would I get? I would get something like this. Where's the solution? Where those two graphs do what? In fact, it would be right about there. There's four squares, so it's going up by 0.25, 1.25, 1.5, I'd go about 1.8-ish. I'm guessing 1.8-something, but oh, they said to one decimal place, I'll bet you 1.8. Is that okay? Then you can do b and actually get the answer with your graphing calculator. I bet you it's 1.8-something. Graph left side, graph right side, second function calculator intersection, first curve, second curve guess, fine. Where'd they cross? C says sketch this. Actually Emily, I don't see one quarter to the x. I see graph y equals four to the negative x because one quarter is four to the negative one with the effect. That equation is the same as that equation, but I like this because it's this equation reflected horizontally. Instead of going through 2.16, it'll go through negative, change colors, Mr. Dewey, it'll go through negative 2.16, and instead of going through 1,4, it'll go through negative 1,4, and it'll still go through 0,1, it'll look an awful lot like that. Now I can tell you what the solution to this is. It's going to be, if I go straight down, it's going to be negative 1.8-ish or negative whatever this number is because it's a reflection. Were there any others on number four that you were going, before I do number eight? Good. Number eight. Okay. Ryan, look up my friend. Eric, what does every single exponential graph look like? What's the asymptote? You're curving closer and closer and closer to the x-axis. The x-axis is a horizontal line zero high. The asymptote of every single exponential graph before we move it is that. How has this graph been moved? Can you guys see? How has this graph been moved? What do you think? Two what and one what? Two right. So here's this horizontal line y equals zero. If I move it two to the right, did that change the horizontal line y equals zero? If I move it one up, ah, it's going to get one higher, Ryan. If I move this graph one up, which is what that means, instead of y equals zero, it's going to be y equals one. Okay. There's a vertical expansion by a factor of four. Then move it to right, then move it one up. We're going to spend a day graphing these by hand, but I usually do that towards the very, very end of the unit. Right now, you know what you really need to know? Absolutely. How does every single exponential graph look like, Eric? Which means it has a domain of all reels, a range y greater than zero. It has a horizontal asymptote y equals zero, x intercept of zero comma one, no y intercept. Sorry, y intercept of zero comma one, no x intercept. Having said that, that leads me wonderfully into the next lesson. Lesson four. Oh, Mitsu, today is like Christmas. I get to give you an entirely brand new function. Woman number one says this in lesson three, in the last lesson in question 6D, which I did not assign, you were asked to find the inverse of that. This is a real problem. How do I find an inverse? There's two steps. The first is switch and then get the y by itself. The first step, switching the x and y is very, very easy, but where is the y now sitting when I do that first step, Spencer? How do I get an exponent by itself? You have no mathematical operation that will get an exponent down to ground level where you can do math with it. In fact, to do that, today, we are going to have to define an entirely brand new mathematical operation. And the mathematical operation is called the logarithm or the log for short. A logarithmic function is defined as the inverse of an exponential function. That's its definition. Remember to find an inverse, we switch x and y, we solve for y, but for the inverse of an exponential, it's tough to get the y by itself because it ends up as an exponent. So what we're going to do is we're going to tell you that if you know one equation, Haley, you know two equations. If you know x equals two to the y, you automatically know, and here's how you read this, that y equals the log base two of x. What's my base of my exponent? Two. What's my base of my logarithm? Two. And you write the base as a little subscripted right below the g and to the right of the g just down there. If you know one, you know the other. In fact, if you have any base b, this is the definition of a logarithm, and these are its restrictions. Y equals the log base b of x, Eric, what does every single exponential graph look like? Could you hold that pose for a second? Now, that's the exponential. What's your range? What's the range of that, folks? What's the range, the range of that? Not all reals. What's the range? Y greater than zero, and if logs are inverses, your range should become the log's domain. All your y's have to be bigger than zero for an exponent. All your x's have to be to the right of zero for the exponential, see it? You can't take the log of a negative number. Oh, for the exponential, thank you Eric, you can put your hands down. For the exponential, the base was b, for the log, the base was b. We had a couple of restrictions on the base of the exponential. We said that the base couldn't be negative. We also said the base couldn't be one, because what's one cubed? What's one to the fourth? What's one to the fifth? Is that going to give me this shape? So we're going to say, you know what? One is the weird base. We're going to, and you notice base can't be one, and base can't be negative. Now that's this definition. Let's talk about what it looks like, and then we'll talk about an easier way to remember how it works. Says comparing the graphs, what we're going to do is we're going to graph two to the x, and this I can do because I know my exponents. I'm going to start with zero, work my way to the right. If I put a zero in for x, what's two to the zero? What's anything to the zero power? One. In fact, two to the x is going to go through zero comma one. If I put a one there, what's two to the one power? It's going to go through one comma two. If I put a two in for x, I'll get y equals two squared. If x is two, what's y? Four. And if I put a three in for x, I'll get y equals two cubed. If x is three, what's y? Eight. One, two, three, four, five, six, seven, eight. If x is four, y is 16, which is not going to fit on my graph, so yeah. Look at the graph. We're putting the three in for the x. We're not saying two to this power, right? I'm glad you made that mistake. It's a sloppy one, but it happens all the time. So since you've done, don't do it, and everybody don't do that.  And let's put a negative one right there. What's two to the negative one? That's a one half. You guys can't read that.  This is the graph that Eric has made famous. Okay? There it is in all its glory. There's an exponential graph. Range greater than zero, horizontal asymptote, y equals zero, domain, all reels. How do I find an inverse? So I'll change colors. The inverse is going to go through instead of zero one, one zero. And instead of one over two up, two over one up. And instead of two over four up, four over two up. And this point here, which is three comma eight, is going to become eight, one, two, three, four, five, six, seven, eight, comma three. And this point right here, which was negative one comma point five, is going to be point five comma negative one. It's going to be down there. You with me, Brett? I'm just switching the x and y's around on my key points. Oh, and instead of a horizontal asymptote, what's the inverse of horizontal? The invertical asymptote. In fact, the graph's going to look like this. That's the log graph. And I can do my double check that this is an inverse by going through the line y equals x. And when I do that, is it reflected about the, yes it is, okay? Derek, what does every single exponential graph look like? Okay? What does every log graph look like, the inverse of that? And that's how I find it. I don't actually have it memorized, you put your hands up, at the tip of my fingers. Well, okay, that's a fib. After teaching it for 10 years, I do. But I generally tell kids, don't freak. I'll show you what I mean. It says, complete the table below. I have yelled at you and said, you need to know the domain of an exponential graph. Is this an exponential graph? Where is the x sitting in an exponent? So it is an exponential graph. What's the domain of every exponential? When Eric did, we won't make Eric do his thing, but visualize Eric doing his thing. What was the domain? All reels. How do I find an inverse? Switch the x and y around. So you know what? What's the range of the log graph? All reels. Look at the red graph that I drew. Does it go all the way down to negative infinity? Yes. Is it getting bigger and big, higher and higher? It's leveling out, but it's never quite going to level out. It's always going to keep climbing. It's eventually going to get to positive infinity. Oh, what was the range of my exponential graph? What's my domain of my log? We're literally switching the x and ys around. Y greater than zero. X greater than zero. This next one, Sandaly, is a trick question. What's the x-intercept of the exponential graph? It's a trick question. Is none. What's the y-intercept of the log graph? No, the y-intercept of the log graph. No, y-intercept. What was the x-intercept of the exponential? Is none. What's the y-intercept of the log? If it's truly the inverse and we switch x and y around, what's the answer? Is none. Look at my graph. Does it ever touch the y-axis in red? No. Gets closer and closer and closer and closer and closer.  What was the y-intercept of the exponential? What, comma, what? Zero, comma, one. What's the x-intercept of the log, what, comma, what? Ah. What was the asymptote, Trevor, of the exponential graph? I caught you zoning out. I know. You ready with me, Beth? See the blue graph? See how it's getting closer and closer and closer to the x-axis? That's the asymptote. Now, what's the equation of the x-axis? It's a horizontal line. How high is the horizontal line? No. What's the x-axis? How high is the x-axis? Zero. OK, so you ready with me? You ready with me? You OK? Throw it so or something? You OK? You good? Happy joy? OK, imagine switching a 3-pointer. Now you're back here. Ready, ready, ready, ready? OK. Trevor, see the blue graph? It has an asymptote. Can you see how it's getting closer and closer and closer to a height of zero? See it? It's getting closer and closer to the x-axis. What's the equation of the x-axis? How high is the x-axis? What did I say the height was? So the asymptote of an exponential is that. So you ready, Trevor? What's the asymptote of the log graph if it's the inverse? How do you find an inverse? How do you find an inverse? So what's the asymptote? That wasn't so hard, but looking over at Brett is not as smart as you think. I assume you're looking over at Mitsu because you wouldn't be in the way, right? OK. Do you see how we got all that? You can memorize both. I really memorize the exponential inside and out. I know everything about it. And then I say, oh, the log's the inverse. I can kind of figure out whatever I need to. I memorize some stuff about the log. But if I forget, if I panic, I got my fall back. It's the inverse of the exponential, which means switch the x and y's around, which means think about it. You know the domain and the range and the x-intercepts and the y-intercepts and the asymptotes if you know the exponentials. Turn the page. What is a logarithm, you ask? What is a logarithm, you ask? What is a logarithm, you ask? Kara, I am so glad you asked. Put your pencils down and see if you can figure it out. I'm going to tell you that the log base 5 of 25 is 2. I'm going to tell you that the log base 6 of 36 is 2. Don't worry about your log button. Jolie, back here. Put the calculator away. This is important and crucial. Cut it out. I'm going to tell you that the log base 2 of 8 is 3. I'm going to tell you that the log base, oh heck, 3 of 81 is 4. Are you spotting the pattern? Are you seeing what I'm doing? Are you also cluing in why a few days ago I said it was worth memorizing certain numbers that we all wrote on a certain blank page? So see if you can figure this out. What's the log base 7 of 49? So when I write this, what I'm really saying, Carly, is 7 to what power equals 49? What's the log base 2 of 1 over 16? First of all, forget the 1 over. What would the log base 2 of 16 be? What about 1 over? Oh yeah, there was that elevator thing. Yeah. Oh, what's the log base 8 of 8? Now this one is so obvious that it's tough. 8 to what power equals 8? What's the log base 8 of 1? In fact, these two become general rules. We say this. The log base A of 8, what's the log base any number of itself? A to what power equals A? 1. And what's the log base any number of 1? A to what power equals 0? 5 to what power equals the square root of 5? Did Mr. Duick say that it was worth remembering that square root was the same as an exponent? Why, yes, he did. Did Mr. Duick say that, or here's another one. Log base 6 of the cube root of 6 equals what? What was 1 half? I gave the answer away. Shoot. What was square root as an exponent? And the answer is, really, I don't see square root. I see this. And what you're really saying is 5 to what power equals 5 to the 1 half? 5 to the 1 half equals 5 to the 1 half? So without me rewriting this one, what's cube root as an exponent? This is really 6 to the what equals 6 to the 1 third. Turns out the answer to a log is an exponent, which is why we spent so much time on exponents starting out of this unit, OK? If you know one, you know the other. If you know the log base 6 of 216 equals 3, you automatically know this as well, yes? So if you know the log base a of b equals c, spot the pattern. Can you tell me what I automatically know? What to the power of what equals what? See if you can fill in the blanks, but look at the example above with numbers that you know the answer for. A to the power of c equals b, OK? Remember earlier, we asked, what would the inverse be? And we ended up with x equals b to the y. We ended up with that. Could you go this way and write it as a logarithm? Write that using this pattern as a log. By the way, have you noticed your base is your base? Your base is the base is the base is the base is the base. So that's one thing you're going to memorize in this pattern of where was the y? Exponent, where is it now by itself on ground level? This was the whole issue we were trying to deal with. How the heck do we get something from up there to here? Pick your pencils up. So here is our pattern over here. We're going to write our template. And from now on in this unit, if you know one, you know both. If you know, oh heck, w to the x equals y, you automatically know that the log base w of y equals x. You'll memorize this pattern as a general pattern. But the memorization should take care of itself. But I'm telling you right now, this is going to be the first question on your test, multiple choice. I'm not even saying I like this question. I'm saying I'm so engaged to this question that we've been in a lifelong, committed relationship. It's going to be on your test. It's going to be multiple choice. So it's asking us to practice. Here's the log form. Here's the exponential form. Let's try this one right here. The log base 2 of 1, what's that? 0, because 2 to the 0 equals 1. If you know one version, you know the other version. If you know the log, you know the exponent. If you know the exponent, you know the log. Let's try this one here. The log base 2 of 1 quarter equals what? 2 to the negative 2 equals 1. 2 to the negative 2 equals 1 quarter. The log answers are the exponents. And since the domain of the log function, the x's were greater than 0. As it turns out, you can't take the log of a negative. You'll get an error on your calculator once I show you how to use your calculators. What are the characteristics? So this is, again, one of those little pages that you might want to dog ear or footnote or bookmark. It says this. Here's my exponential. The log is the inverse. So the exponential has a y-intercept of 1. The log has an x-intercept of 1. The exponential has no x-intercept. Sandaly, the log has no y-intercept. Trevor, the exponential has the x-axis y equals 0 as an asymptote. The log graph has the y-axis, which as an equation, is x equals 0 as a vertical asymptote. And then Vlad, here's the set notation you were asking me about. We'll make a little note. This vertical line right here means such that or belongs to is a member of, belongs to the subset of. There's all sorts of ways to say it. But I read this statement as saying the domain is all x's, such that they're positive. And it's all of the real numbers above 0. So not just 1, 2, 3, but decimals as well. All y's, such that y's are all real numbers. Now, you don't need to write that. All you need to write in your answer this year is that part and that part. I'm not worried about the funky set notation. If you've got a mind that they did it, it is uber-nerdly cool. And if you know 1, you know both. There's the restrictions on the log. x is positive, the base is positive, the base can't be 1. Turn the page. Example 1 says convert each of the following from logarithmic form to exponential form. So here's the log equation. What's the exponential version of this one? What to the power of what equals what? I'll give you a hint.  What's my base? What's my base? To the power of what? You have the template on the previous page, so look at it. Figure it out. 7 to the 4 equals x. I want you to notice, Dominique, what we just did. Where was the x originally inside a log where I don't know how to solve for it? Did we just get the x by itself? In fact, if we went 7 to the fourth on our calculator, we would have just solved this equation. I'm not going to bother. It's some big number. What to the power of what equals what? I'll give you a hint. Your base is your base is your base. 10 to the third equals 1,000. Now this one, you can tell whether or not you've got it right, because if you wrote 10 to the 1,000 power equals 3, you'd be saying, that's dumb math. I'm not going to give you one like this.  One like c. Something purely algebraic, and I'm going to say write that as an exponent. But I'll still give you the same hint.  What to the power of what equals what? T to the m equals b. D, what to the power of what equals what? Well, my base is my base is my, OK, to the power of what? Now I would have no problem if you wrote that equals a. I'll tell you how the answer, though, would look. Do I have a power to a power right here? What's my rule for power to a power? I almost certainly guarantee because they're typing and it's hard to do a subscript on a subscript when you're typing, they probably just multiply them together and write it as bd, because that's what really would happen. Before I can do e, I need to get the log by itself. What's the for doing to the log in front mathematically? Do you think from all your years of math experience, when there's no sign there, what are we doing? Timesing, how to move the for over. First thing I'm going to do is this. 5 over 4 equals log base b of 6.  By the way, this is also now an equation we could actually solve. What kind of an exponent do I have here? Fractional? Didn't we a couple of days ago learn a trick to get rid of fractional exponents? It was we did something to both sides. Both sides to the what? Better know this for the test. Didn't we go both sides to the reciprocal power? If we went both sides to the 4 fifths, we could solve for b, nod. Yes, we did. Lesson two. Two lessons ago, we're now in lesson four. Example two, not only can I write something as an exponent if I know the log, I can also write something as a log if I know the exponent. So if 4 to the third equals 64, I'm going to get the log base what of what equals what? Base 4. Don't say log 4. If you say log 4, I'm going to assume it's here not a base. So log base 4 of 64 equals 3. 2 to the negative 3 equals 1 over 8. Sorry, log base what of what equals what? And if I have something complicated inside the log, put it in brackets so I know that it's all inside the log. So if it's a fraction or some kind of expression, that also means that 2 to the negative 3 equals 1 eighth, which is true. A and B, I will not give you as a question on a test because you can figure out if your answer makes sense by looking at the numbers. I'm going to give you an algebraic one like C. Log base what of what equals what? I guarantee you the first question on your test is either going to be C or C. I'm either going to say here's an exponent right out as a log, or here's a log right out as an exponent, or perhaps both. I'll have one of each. There will be multiple choice questions. In fact, I'll even tell you what the answers would be. For this one here, I would also have log base E of D equals F. And then I'd probably just mix the letters up in different locations. There's what the answers would be. If you're getting that question wrong, you're flunking math 12. I'd almost be willing to put down $500. This is the unit where either kids say I've made the adjustment to the pace of math 12 or where they say I'm dropping the course. So keep up with the homework. A couple of trickier ones. D. Oh, my base is my base is my base is my base. Log base X of 5 equals 2y. E.   But log base what? 2x plus 4, and I'm going to put that in brackets because it's a pretty ugly one. A equals negative 1. Sometimes instead of asking you to rewrite it, we'll actually ask you to evaluate something. So this says solve for y. Is the y by itself already? Yeah, good. Then what is the log base 3 of 81? That's the answer, apparently. 3 to what power equals 81? Log base 5 of the square root of 125. No, relax. I would rewrite this first as the log base 5. See that number 125? They didn't pick that at random. I think that 125 is actually a 5, my base. How can I write 125 as a 5? Sandaling. Write that down, and then don't write this next bit down. Hold your pencils for a second. Technically, it's all inside a square root, except I don't want to write the square root, which is why I said don't write this down. Square root is also an exponent. Is it not? What? OK. I'm going to write this as log base 5 of 5 to the 3 to the 1 half. How does that help me? See the 3? See the 1 half? Is that a power to a power? What do I do with the exponents then? Well, I multiply. This is actually going to become y equals the log base 5 of 5. Remember, multiplying fractions was the easiest. It was top times top, bottom times bottom, 3 times 1 over 1 times 2. And it's now so easy that it's tough. 5 to what power equals 5 to the power of 3 over 2? Doesn't 5 to the 3 over 2 equal 5 to the 3 over 2? See what I mean by so easy that this one, kids always flip. No, actually, it's too easy. I'm pretty sure it's that. 5 to the 3 halves equals 5 to the 3 halves. See, I can't do see as is. I need to get the log by itself. How would I get the log by itself? So I'm going to have y over 2 equals the log base 8 of 512. 8 to what power equals 512? I don't know. Let's see if I can reason my way there. Is the answer 8 squared? Is 8 squared 512? Kara, what is 8 squared? 8 times 8. What is 8 times 8? That's OK, ready? What is 8 squared? Louder. Yeah, louder. What is 8 squared? So I know it's not 8 squared. I wonder if 64 times 8 is 512. Well, what's 60 times 8 using a nice close round number? 480, is that close to 512? You know what? If they made this question work out evenly, and I'm sure they did because teachers tend to do that, I'm pretty sure it's got to be to the third power. In other words, I think this whole thing here has to work out to a 3. This whole thing here has to work out to a 3 because the log base 8 of 512 is 3. What divided by 2 is 3? What's the value of the log base B of 1? Because B to the 0 equals 1. What's the value of the log base B of B? Because B to the 1 equals B to the 1. Don't write this down, but what's the value of the log base B of B to the 5th? B to what power equals B to the 5th power? 5. In fact, you know what? Let's generalize. Write this one down. What's the value of log base B of B to the nth power? The log base B, sorry, what log base 6 of 6 squared is? Just the 2. Nearly done, a few more. Says evaluate. What's the log base 4 of 64? 4 to what equals 64? 3. What's the log base 2 of 32? Sorry, what's the log base 2 of 1 over 32? Well, I would handle it by saying, what's the log base 2 of 32? Sorry, do you know what 2 to what power equals 32? So since you guessed, everybody, put your hands up. Figures up. Start closed. 2, 4, 8. What comes after 8? What comes after 8? Sorry, what comes after 16? So 32 is 2 to what power? I will feel free, Eric, to ask you any power of 2 up to here. I figure I can ask you to double all the way up to 2 to the 9th. 128, 256, 512, 1,024. Really, 2,048 should be fair game, too. But whatever. I'll be using lots of powers of 2. And I told you your test will have a non-calculator section. So I just showed you how you can do the 2s, fingers. So if I hear you correctly, you said 32 is 2 to the 5th. What about 1 over 32? Right, elevator, right? 2 to the negative 5 equals 1 over 32. By the way, if I gave this as a multiple choice question, Asar, do you think I'd also have 1 over 5 as an answer to pick from? 1 over negative 5 as an answer to pick from? 4, 1 quarter as an answer to pick from? Actually, you know what? I probably have a 16 as an answer to pick from because a lot of kids divide for... Oh, 32 divided by 16! No, that's not exponents. C. Well, where do logs fall under the bed mass rules? Because they're an exponent, they fall under the E for exponent. Do the exponent first. What's up in the exponent? I don't see a log base 5 of 25, because I know what that is. You know the log base 5 of 25 is? I see this question actually as that. Suddenly now it's math. Six, grade six, I think. Five squared, five times five. D, A to the log base A of A. Do the exponent first. What is the log base A of A? This is just A to the 1, which is just plain old A. Sample six. It says, find the inverse of the following equations and answer in the form y equals blank, get the y by itself. Okay, Kara, how do I find an inverse? So let's do that for this first one. X equals the log base 3 of y. Now they want me to get the y by itself. I can't if the y is inside a log, but if I know one equation, I know the other equation. If I know a log, I know the exponent. If I know the exponent, I know the log. So in the desperate attempt to succeed, I'm gonna rewrite this as an exponent. What to the power of what equals what? Is the y by itself? So have I followed the instructions? I found the inverse and got the y by itself. There you go. Is the y on ground level? Yep. By the way, normally they'd write y equals 3 of the x instead of 3 of the x equals y, but whatever. Isabella, how do I find an inverse again? So for b, I would go, okay, hot shot. X equals 8 to the y. Get the y by itself. Now, where is the y sitting, Isabella? It's exploding. I can't do anything. Oh, wait, wait, wait, wait. If I know one, I know the other. Let's rewrite this as a log. Log base what of what equals what? Oh, the base is the base is the base. Log base 8 of equals, equals Justin. Is the y by itself now? Oh, that was quite convenient, actually. Warmup number three, blah, blah, blah. Let's just jump to the examples. Here's what they want you to realize in example A. Before you can rewrite the exponent as a log, you have to get the exponent by itself. What's in front of the exponent in A? A 2. Oh, now you got me yawning. A 2. Can I go two times three is six? Everybody say no. Wasn't vehement enough. Can I go two times three equals six? Everybody say no. Not good enough. Can I go two times three equals six? Everybody say no. No. Okay, okay, get your hand away from your mouth. Did you say no? I couldn't see your lips moving. Can I go two times three equals six? Say no. Instead, I'm gonna get the exponent by itself. What's happening between the two and the bracket? I wanna move it over then. I'm gonna start out quickly rewriting this equation as follows. Now, what to the power of what equals what? Sorry, log base what of what equals what? Log base three of y over two equals x. Skip B, let's do C. Ashley, is the exponent, is the power by itself already? In C? Say yes. Don't you want it? I caught you mid-yawn, sorry. Yes, yes, yes, yes. So I don't need to divide to get it by, it's already by itself. Let's rewrite this as a logarithm. Log base what of what equals what? Log base D. Crack it, relax. Yeah, probably not, but it's gonna get tough to tell whether the y is also part of the log or not. I just, as soon as there's more than one thing in the log, I usually put it in brackets if they have it. Safety line. D, y equals three over two to the 10x. Okay, let's get the 10 to the x by itself. Fair enough? What's the three doing to the 10? Times thing, how will I move it over? Fine. What's the two doing to the 10 dividing? How will I move it over? Multiply. In fact, I'm gonna get this. Change colors, what you're doing. Two y over three equals 10 to the x, which means the log base what? 10 of what equals x. Got me yawning again. Try e on your own. See, I passed that right on to you, Kara. Try e on your own. Algebraic, get the power by itself first. Log base x, log base s of t over r equals p. Yeah, yeah? Excellent. Can also go from log to exponential. And then we're done. What to the power of what equals what? I'll give you a hint.  Seven to the power of. x equals, now y over three. In the instructions, they want me to get the y by itself. So instead of writing over three, instead of dividing by three, what's that the same as doing on this side? I'm gonna put a three there and I'll put that in brackets and there's my mzing by three.  Two steps at once, not the end of the world. B, this is gonna be 10 to the x equals y over four. You know what, instead of over four, I'll times by four over here. C, five to the power of x equals seven. Wait a minute, how would I get the y by itself on this line? I'm gonna go boom, boom, multiply by one seven. I could divide by seven and I wouldn't take marks off if you did. I'm just worried if I go divided by seven, I'll think the exponent is on the seven as well and I don't wanna take that chance. So I usually put it in front as a fraction just to be paranoid. E to the x equals y over five. How would I get rid of the over five? Multiply homework time. One all is good, skip two, three all is good. Four all is good. Five all is good and six all is good. So right now I've gone one, two, three, four, five, six, but I've skipped number two. So really one, three, four, five, six. Eight is good, nine is good. Yes, I'm giving you lots of practice. 10, A, D, C, F, eight, sorry, I skipped 11, 12, yeah, 12 is good, 14 is good, 16 is good. There it is, the mighty logarithm.


Hi guys and welcome back to another episode of our Pokemon VGC 2019 Battle Series we are here back again, it is Wednesday and we are playing this Dialga-Kai-Oga team. Now we played a couple of matches this week already and we played this team all last week it's been great fun so far. The last couple of days we've only been able to manage one game per day because they've taken so long so hopefully we can squeeze two games in today, be a lot of fun as always but I hope you've been enjoying the episode so far with this team and like I keep saying, hopefully we can bring the bronze on and the zerkertree a bit more. Now we brought the bronze on and the last couple of games have done a lot of work and it's been really good to see but zerkertree is the one missing out so hopefully we can bring it a bit more as we got into the latter part of this week before we end up with this team. That further ado though let's hop on to the ladder as always if you enjoy this sort of content please make sure to leave a like on this video it really does help the channel and the video itself and make sure you do subscribe to the channel so you don't miss any of the daily battle series that we do have coming out as well as our guides and all of our other content so across my vision too can we finally get you on this episode which would be really good so yeah and all honestly though it's been an absolutely incredible call to play around with I've really like really enjoyed it it's been a lot of fun and I think it's been really successful as well it's had a really good run on the channel for a lot of wins and our win rate is like crazy so what did we lose two last week played like 10 matches 11 matches so 11 like 9 to so we're like 11-2 at the minute with this team which is pretty good pretty good and hopefully it can just continue and every time I say this feels like we're jinxing ourselves but it's gonna be really nice to finish up with this weekend it's been a nice call to play and it's definitely one I would consider playing around with and taking to tournament because it's just so solid like really solid and if we can't find the opponent as well by the way guys I'll let I'll just cut to when we do find one and like magic there we go we got Springs VGC 1760 rated player so a really high rated opponent and we'll get straight into team so first opponent today playing a team of Sogaleo, Kyle, Cortana, Toxtokrog, Koko and Hydraegon so Hydraegon very nice to see that Pokemon making its appearance back into this format one of my favorite dragons and has done so well in previous formats it does suffer a little bit when it's up against Zernius but against this sort of team that we're playing it probably does and has a really nice time so we're gonna find it very difficult to deal with it because we've not got too much to hit it very super effectively like at all and so dealing with it is gonna be extremely difficult I feel like we're gonna be struggling against it right from the get-go honestly and what can we do so trick room here for us is gonna be very good Sogaleo could have trick room so we need to be careful of that the Toxtokrog is gonna be another thing that we need to watch out for and be very careful of before and I'm gonna bring Incineroar and I'm gonna bring I definitely need Incineroar for the Intimidate support it's gonna gotta be a little bit careful against a lot of this this team I think maybe our best option in this kind of against this sort of team is to kind of just use brute force and bring Zerkatry and Moongus to kind of try and get through the best part of this team he's dialogue to get a trick room I really feel like my opponent brings the the Hydra again it makes a lot of sense against what we've got we've got no fairy type attack in our team we've got no dragon type attack in our team we've got no ice type attack we've got no fighting we've got no way to hit it super effectively so our only real way to deal with this the Hydra again is real brute force now we could have went down the Kyogre route but then you look at the team it's got Cortana they got Toxtokrog they're really set up well against that Pokemon so going down that brute force route is not gonna be the best way to go I don't feel so it does make it a little bit more difficult and we are gonna see the Toxtokrog Kyogre come out from my opponent now we've got a couple of options here we can trade fake outs it's a little bit dangerous because you could lose Insino on Insino is quite important for us there in this game there's also the aspect of fend that we could see fend Hydra vortex or fend waterspout into the the Dialga which wouldn't be too far fetched I think to be safe I'm gonna bring in Amungus and I'm just gonna I'm just gonna protect I don't see fend coming out this turn with us having fake out pressure on the field I think it's more likely to be safe fake out than anything else so we'll bring in Amungus at least with Amungus the next turn we'll be able to guarantee our trick room up as well and then we'll protect the Dialga just make sure that we're not taking too much unnecessary damages first and there's a fake out from the Toxtokrog into Dialga we could have just traded fake outs this turn but deciding not to and thinking more about next turn when the Toxtokrogs able to potentially fire off one of those low kicks and we want to be able to rage part of that away and and set our trick room up and kind of start getting a little bit momentum in this match another waterspout from the Kyogre will probably just proc the berry on Amungus and if we see poison jab there which may be likely to Amungus will be in a nice position this next turn as you see there another waterspout yeah and there's the berry proc and we'll pull in whatever attack from this Toxtokrog is probably gonna go for a low kick I would imagine poison jab yeah making sense because knowing about the rage powder here and doing some nice damage but we are able to get this trick room up and that does put us in a decent position going into this next turn where we can spoil the Toxtokrog and we can thunder the Kyogre just to get some decent damage onto that we've got to be a bit aware of what my opponent has in the back potentially to bring in but there's nothing really that I'm too concerned about coming in like the Hydraegon comes in then we can put it to sleep the next turn we're sitting in a decent position going into this one so Tabogogo gonna come in now get the electric trainer that's just boosting our ability to you see this is the thing now now that's a nice player to cover the spoil there and it was one thing that we just didn't really consider too much of this so nice play for my opponent with the thunder low kick now I'm gonna come out into that I'll go probably be enough to pick up the knockout not quite enough just poking that berry so that's defensive he's really coming in handy for us here and if you need a reminder of what the builds are on this specific team you can go down into the comment section below and in the description sorry and check those out there I think this is probably a good time to get a zerkatree in on the field so I'm gonna do that for a moment and I'm going to power the Tabogogo because the Toxtokrog cannot knock out the Dialga this turn but we can knock out the Tabogogo if it targets down Dialga again then that does mean that we okay Toxtokrog switching out that's fine so we're out coming in that's also not too bad Cocoa protected so we'll be able to get the Cocoa the next turn in tail blow my opponent hasn't got any good switching other than the Kyogre for that slot and we've still got two turns of trick room so we've got at least this turn to get the tail blow up and she get the Cocoa here maybe we see the Toxtokrog come back in with that solve vessel be able to sock up the earth power pretty nicely whereas Cocoa doesn't want to take it yeah it's switching out Kyogre coming in maybe Toxtokrog yeah like we thought and it's got a fake out support going into the next turn so Galeo just gonna protect here which is fine but we'll be able to get this tail blow off with the Zerkatree and we can get a moongus back on the field this next turn if you would like to and then you still gotta be a bit wary about this over there I will take the Toxtokrog down to focus as revealed that's interesting item on it not the assault vest the ring does stop as well preventing any sort of them so I think what we could potentially do here is we want to preserve the Algar as best we can and I'm gonna actually just protect this is where Kyogre comes back in I think I kind of want to double up into the the Sogaleo slot here with a Thunderbolt then a Thunder but I don't want to leave the Toxtokrog and check but it definitely goes for a fake out I'm pretty sure I think what I'm gonna do is Thunderbolt the Sogaleo and I'm gonna I'm gonna switch Incinero into the Algar so I've got fake out of my on this next turn and that means if the Kyogre comes back in and can really disrupt that slot and we could see the double up into Zerkatree but it does leave the Algar open too yeah there's the there and I'm gonna see since you'll strike yeah it's gonna be into the Zerkatree slot I'm hoping that this is just if this is life orbits just enough to pick up the oh it's into Incinero huh that's interesting I mentioned send back to normal okay it's not likely that the Toxtokrog's got protect here so what I'm gonna do I think is go for the Toxtokrog just Toxtokrog so I think you just sack everything at this point really I really do I think you sack most things to alleviate this I need to attack the Sogaleo for sure and I'm in you turn out on the Toxtokrog I don't think a double up takes down Zerkatree especially once we've intimidated the Toxtokrog so they're gonna protect yeah I think you just like sack this Pokemon this turn yeah there's a poison jab it'll be into the Zerkatree for sure yeah no poison which is nice okay now that no I think we bring in the movies or do we bring in Dialga hmm because Kyogre comes back in now for sure kind of means for going Zerkatree though and then gone for the second Zerkatree that's the one thing I'm not really keen on doing and but at least we haven't got the Hydra again to think about here so it's what do we bring in Kyogre comes in and your water spout yeah I think we need to bring in Dialga maybe protect it maybe sack Zerkatree or switch in a moongus so we can get the trick room up the next turn because now the Toxtokrog's gone like everything on our team is okay so they're actually trained disappearing Kyogre's definitely coming in I think and I think it's definitely scarfed yeah we sack Zerkatree here and then bring in in the center all thing that's the only thing we can do is yeah Thunderbolt I'm gonna Thunderbolt the Sugalail though and I'm just gonna protect Dialga I don't think there's no way we get the Thunderbolt off unless you double into the Dialga for some reason but I just don't see that yeah the water spark it just makes way more sense to do that yeah and even now it's still really difficult for us to do anything because the superpower comes out but it sends seal strike every time I wonder if it's not got super power yeah I don't think it's got super power really don't and yeah so we're getting Cineruin we can if it's not got super power then it can't deal with like we can can easily close this game up it makes a lot of sense if you've got if you've got super power you go for it here but the sense seal strike she's not gonna be enough to do to damage the Dialga on the Incinero we could be wrong we could see it here right now that causes got it you just predicting different things to come in on different slots oh Dialga hangs on skin of its teeth oh my gosh okay that's huge for us that is really huge for us and now the Sugalail is under a lot of pressure so is the Kayoga and I think the Kayoga switches out to Koko if I'm honest but I'm gonna knock off the Sugalail and I'm gonna thunder the Kayoga opponent knows we've got thunder as well can't believe the survival there that is huge knock off should do good damage and get rid of something we see oh it's a Z move the Sugalail has got the Z move that's pretty big that thunder is nice damage what's about gonna come out take down the Dialga probably take down Incinero should proc a berry I think we're just on verge of berry so we might be able to take if it's a Z move though we we are not taking the Z move if it's the Phytinium Z we're not taking it oh it's trick room nice nice okay that's cool it's a really nice option there with the Sugalail sometimes to see carry protect why guard protect trick room makes a lot of sense though and the Kayoga in a bit of a tricky spot though because the water spots just not doing enough damage at this point even though there is a superpower there from the Sugalail the Amungus is still putting on a lot of pressure on majority of the field and you're gonna have to see really the type of Koko come in here for the Kayoga I would imagine that would be the best thing to do and to prevent any sleep spam yeah I wonder if we see the Z move now here it is oh it's a Stelium Z nice okay that's cool I mean Amungus probably takes this it's minus one there just massive props really great team yeah that's putting Amungus right out of reach really grass not gonna not good to do near enough and Kayoga is able to gonna be able to come in right now and and just clean up both Pokemon unfortunately so we can't do too much about it especially with the rain up and when does the rain finish actually ends now so yeah there's no chance we needed the rain to end turn after the Kayoga came in but it's not gonna happen it's not gonna happen and we can rage powder I mean but Amungus I don't even think takes a thunderbolt from this rage Sugalail just too strong and thunder boom boom boom there we go that pretty much locks it up for my opponent you're gonna super far do enough to the Incineroar but I mean even at this point we've got no way to come back so we so worried about the Hydragan at the start of the match and for good reason as well like we don't have a way to deal with Hydragan but the rest of the team the team that my opponent brought has done super well Sugalail is so bulky as well make it very difficult for us to to move around then a trick room pull at the end there really kind of pulling it back in for them so Kayoga gonna come back in now full force with this let's tap Koko get the rain up perfectly timed and we will smash that forfeit button but very good game to my opponent and like I say really nice team composition really like it Sugalail Kayoga one of my favorite like combinations in restrict format that we've had since the Sun series and it's such a nice team to see built around that restricted pair so a very good game to my opponent pleasure to play and we'll move on to our next game guys really doesn't take too long to find opponent but Springs VGC doing all the good stuff there come on come on come on let's go right there we go on to the next one like I say hopefully it doesn't take too long if it does we'll cut to it and come back to it next opponent playing from Japan 1450 rain playing a team of Zornius crowd on Tabukoko in Sinoro Zatu and Amugus Zatu something you don't see too often but an interesting pig primarily gonna be tailwind I imagine from that Zatu I'm presuming it gets it and then you got the Amugus there's a trick room check I guess from the other side of things and bronze I'm gonna be really good here especially with intimidate support and the rain support from our side especially cuz that Amugus is the only thing that's kind of threatening it and the trick room putting it to sleep so I will bring that call that's what we want to bring as our last one so we're gonna lead off with the bronze on in Sinoro Kyogre in the back and I think Darga could be good Amugus on probably not gonna be too useful Darga let's go with Darga yeah cuz Darga's just a beast isn't it so we're locked in ready to go and hopefully my opponent doesn't take too much longer to decide their team Zatu though it's kind of yeah it's an interesting one I'm sure it gets magic bounce is it magic bounce I think it's ability something like that I know it's similar to yeah it's something like I'm not too sure but you can't you can't taunt and things like that cuz it just bounces back it's the one that Absol mega Absol has I think mega Absol am I just confusing it with something else I don't know I don't know but yeah we'll not be taunting it cuz we don't have taunt on our team or anything like that so it's fine just generic trick room set up and try and blast through things which has been pretty difficult to do with just Kyogre recently but hopefully we can pull out a win to end up today which would be very nice to sing grow down and have a go come up my opponent as we need off with the bronze on and in Sinoro now the taba Coco lead here indicates that that Coco probably has taunt I would imagine and I'm quite happy faking it out and trick rooming to get around that taunt if I'm like completely honest with you yeah I don't really care about like if we lose in Sinoro we lose in Sinoro we get Kyogre and we've got skillswap so we don't even care about having to reposition Kyogre get it out getting it in again it makes it so much easier we just need to get this trick room up and then we can totally smash smash things mungas coming in which is still don't mind because the electric terrains are bish bash bosh yeah so it's fine and we can totally just keep in Sinoro out here and flab let's see mungas this next turn gyro ball the taba Coco I guess that should be super nice because you can try and put bronze on to sleep as much as you like but it's not going to work it's not going to work there lad you gotta imagine I'm intimidated in the Sun flab let's it's gonna be more than enough to take down a mungas it should be okay mungas gonna switch trade back out I even want to entertain anything I'm staying on the field and taking a flab let's gonna see the in Sinoro come out could maybe play a little bit more a little bit better go for you turn here to see the cock would just protect all right the damage on to this in Sinoro regardless anyway so that's why it's a white in it any calf okay I kind of want to I want it I want to you turn out with her in Sinoro I'm gonna bring the algorithm I don't know if we'll be able to you turn out or not but if we can it would be so nice but in the same time the taba Coco could probably vote switch out as well which makes it I'm so sorry so sorry it's been a long day and which makes it a bit more difficult for us to get weather control especially bronze on now and not on the field so we'll get the you turn off maybe the sloths it's in around the field she's interesting I think we get do we get Kyler in let's just get Kyler in no you both switch out I did we get Kyler going I reckon they're doubling in on that that bronze on slot so I think it's pretty safe for us to get Kyler on to the field but your opponent is always an unpredictable beast so you can't really account for what you think would make sense always being the thing that makes sense we'll see another you turn for these it's gonna get the groudon back in what's your moongus and a you turn double you turn huh now the groudon comes in it's a slob groudon so yeah I think we just protect with Kyler we get bronze on in as long as we've got trick room turns up I don't know if we have we've been funny and round so we've not got too many turns left okay let's get bronze on there and protect Kyler on this turn and then we can skill swap waterspout on our last turn of trick room and as long as we don't see a fire punch oh I'm sorry I need to stop yawning drink some water it's so bad I've had the worst headache all day today and it's just finally gone and literally woke up with it this morning had it all day and it got really bad I'm not one for taking pink colors unless I really really have to like my grade level and it didn't quite get there but it was still pretty bad it was really disorienting all day but it's finally gone but now it's gone I'm really tired I don't know don't understand it Tabacoco coming back in for the movies this doesn't help us out too much because we want the Tabacoco to be forced out so we can get that electric train back onto the field oh it's the tectonic reach wonder if it's into Dialga I'd feel pretty safe with the Kyler out on the field right now yeah okay that was a nice switch from us if I do not say so myself so we'll skill swap and we'll water spout yeah but I don't know if you get this guys where you get like if you ever wake up in the morning I think it's because I'm doing like a pure keto diet at the minute and some side effects of it are headaches to begin with until your body like really adjusts to getting into ketosis and tiredness as well so probably a combination of both but I'm sorry that I'm doing it on camera when I'm recording I'm just trying to squeeze in matches when I can because it's quite it's quite difficult sometimes to to always be able to to get this many episodes in like you know what I mean around everything else that I do as well like full-time job and stuff like the basic grind of nine to five yeah but it's fine I enjoy doing this and I enjoy doing it for you guys we are gonna get that skill swap off get rid of the ground on surprise okay ground on down a trick I'm gonna end and the electric train gonna disappear this makes a moon is a lot more threatening for us now we need cocoa to go away come back in and get that electric train up I need a mask I need a mask to just hide my own okay we can we can get another trick room up for sure I think what we could do here is probably just a light switch and what's about again it's pretty cheeky but we haven't revealed the allies which yet and I don't think my opponents predicting that this could go way south way quickly for my opponent I mean if they target into the bronzong assuming that Kyogre protects then I mean okay so we just get a forfeit and end up there so I think we would have been able to close that one out but good games my opponent and that is the end of today's episode guys so thank you so much for tuning in and I really do hope you have enjoyed today's episode it's been a lot of fun just realized I only put the music on for today's episode which is a big error on my part but I'll put some nice overlay music on the whole episode so we'll get some Colosseum music on this is some of my favorite ones anyway so I hope you have a great evening look forward to reading the comments about today's episode guys what a great team we played earlier on spring VGC so that was amazing and we'll be back for more miss chief tomorrow so miss chief managed guys take care have a great night and until the next one I'll see you then so bye bye


Okay, we're now in the last group of design patterns object-oriented design patterns and those are the creation of patterns So how do we go about creating instances and the first one? We will discuss is the so-called singleton pattern, which is all about Creating a single Instance of something and that's often the case if you want to control something So you want to make sure there's for example only one end point of a component only one interface that Communicates with other components. You don't want to have this double because they might have access to a restricted resource For example, if your component is in the network They have to listen they have to react to any kind of calls that come at a certain address and at this address. It can only be one Component one interface at a time So if you would have multiple of these controllers They would probably kind of restrict themselves in the excess or you have access to a memory location And only one at a time can read or write that's when you want to Maybe consider the singleton now. There's a lot of stuff going on here, even though it looks very simple and I'll Walk you through that. So first of all You should notice that The instance of a singleton so this class has an instance of itself, but this is a class attribute So there's only one of these for all the classes And even if you don't have an instance of singleton this Attribute exists for the class. So that's for example static in many programming languages The same goes for the method they get instance method that returns a singleton is also a class method it static You can call it even if there is no instance and then the third thing that's going on here is the constructor The constructor of the singleton is private. So In most programming languages, that means you cannot call this which means you cannot Create a new instance by for example doing new singleton This would in most programming languages not be possible because the singleton constructor is private The way this works is that instead you call get instance So for example in programming languages like Java this would look like singleton with a large s dot get instance so you just call it on the class level without creating a a new object and This one will then return you an object and the way it works is simply that this is a class method It checks whether the class attribute instance whether this one is set already If it's not for example, if it's now or undefined again depending on the programming language if it's not then It calls the constructor and it can't it can call the constructor because It's private. It's within the class. So this method is actually able to create a new instance This method is able to call new singleton. So if the instance undefined it calls this method It gets a new instance it returns it if this one is not undefined It simply returns the existing instance. So that way it makes sure that the first time this one is called We are creating one and afterwards. We're just returning it. There's always just one Now I said you might want to consider this and that's because there is a lot of criticism surrounding the singleton pattern So I think it's important to discuss that here I said before most patterns change over time or they might not be appropriate anymore They might be controversial. This is definitely one of them and the reason is Well manifold there are many reasons for this first of all one issue we have here is actually that we are violating the single responsibility because What the singleton pattern does is actually two things it makes sure there's only one instance But it also provides a global access to for example this one to the singleton instance to the controller So it actually has two responsibilities It violates single responsibilities The other thing is that this is very difficult to test it has an own state So the state depends on whether the method has been called or not and Depending on what other methods this might have the state changes over time and stateful Classes are generally not that easy to test. So that's a tricky thing here testing a singleton is hard Instance Access to restricted resources that I have mentioned as one of the reasons why you want to use this might be possible through other ways So it's not ideal to have a singleton if you can avoid it And a suggestion is very often to say well instead of the singleton maybe use something like dependency injection so make sure if you have to have something that has a state just provide the state by By a means of parameters for example to the constructor instead of Implementing a singleton. So it depends a bit on the use case that you're having But it's not always necessary to use this and if you can avoid it You should probably because then you don't have to test this anyhow Okay, so that's the singleton The next and the last pattern will go through in in this module is the factory pattern Which is a common way of as the name suggests Creating instances of the same thing