 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. Oh, oh, oh, oh, let's keep going. 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. So, oh, you know what? 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. Sandaly, are you ready to redeem yourself? 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. Your base is your base is your base is your base. 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. You know what I'm going to give you on your test? 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. Your base is your base is your base is your base. 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. Now, what to the power of what equals what? b to the power of 5 over 4 equals 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. Your base is your base is your base. It's an ugly base. 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. Your base is your base is your base. 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. If I go too fast, you guys okay. 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.