 Well, last day we introduced a brand new function. We introduced the logarithmic function, the log function, the inverse of the exponential and that told us some things about the graph and we're going to look at the graphs in far more detail at another date. But we said right now at this stage, we're doing logs in our head. You are going to want your calculator out. If you don't have a graphing calculator, you need a scientific calculator because we're going to be showing you a few new functions on your calculators later on today in class. But I'll start out as per usual by saying, hey, questions from the homework. What would you like me to go over now as your chance to ask? 16? Absolutely. Any before number 16? 5f I can do. I'm also willing to do 9c if I need to. But we'll see. 5f. First thing that I would do is I would rewrite this as to the 1 half power because that's what square root is. 512, okay. Get up the fingers, 2, 4, 8, 16, 32, 64, 128, 256, 512, 2 to the 9th is 512. So 1 over 512 is 2 to the negative 9. Is that okay so far? So this is really the log base 2 of 2 to the negative 9 over 2. Where the negative 9 over 2, power to a power, top times top, bottom times bottom is a 1 on the bottom of the 9. And this is really saying 2 to what power equals 2 to the negative 9 over 2? I think 2 to the negative 9 over 2 equals 2 to the negative 9 over 2. There's a better way to do that question, a much better way. Next lesson on Wednesday, I'm going to be showing you what I call the log laws. And there's a great shortcut for this one. But right now, what I've just done in the long method is proved why the log law shortcut that I'm going to show you on Wednesday will always work. Any others after number 5 before number 16? Sorry? 12A? I need somebody actually asked me to do one of these because I mentioned it. A bunch of people nodded their heads, but no one's actually told me that. Come on, okay. 9C, I'm so glad you asked. Who would like me to do 9C? Thank you. I love the fact that they stuck 9C in here, Katie, because every time I introduce the log, kids go log happy. Every exponent becomes a log, and that is not the case. The logarithm is the inverse of an exponential equation, like in A, when the x is an exponent, when your variable is an exponent, when the exponent is a variable. Katie, what's the variable here? What's the, sorry, what's the exponent here? What kind of an equation is this? It's not an exponential graph. What kind of an equation is this? It's a quadratic. How do I know? It has a squared. Remember, Kate, we've done this a little bit before, and the reason I do that is because you're going to have a bunch of equations by the end of the year. This is a quadratic, which means there's no need for logs. How do I find an inverse? I switch the x and y around, and then I get the y by itself. How would I get the y by itself here? I think, Katie, I would minus 2 from both sides. Divide by 3, I'd get this, and I'd have a y squared over here, except how do I get rid of a squared? Do I need the logarithm to get rid of a squared? Say no. How do I get rid of a squared? What's the inverse of squaring? In fact, it's going to be this. Oh, but wait a minute. I have to remember plus or minus when I square it both sides. Do you see the difference between A and C? The x is an exponent in A. That's when the inverse is going to end up being a logarithm to get the x or the y down to ground level. C is not an exponential graph. It's a parabola. I'm not going to repeat myself. You'll have to watch it again, you're chatting. I heard somebody 12 something? No, someone asked 12 something. Yeah, what? 12A? What we said last day, Ryan, was from now on, if you have a log, you also have a second equation. Or if you have an exponent, you also have a second equation. We said, if you know one, you know both. This is really 2 cubed equals y over 4. That's the same equation. This is the definition I said you wanted to memorize by the end of today, for sure. I wrote it as an exponent. What is 2 cubed 8? What divided by 4 equals 8? y is 32. Same thing here. I didn't assign this one. Same thing here. What to the power of what equals what? Write it as an exponent. If your variable is inside a log, you get it outside the log by writing it as an exponent. 16. Andrew, there's a much better way to do number 16 with a shortcut that I'll be showing you on Wednesday. But for now, you know what I would do? The same thing as I just did with Nicole. I would realize that if I know this, I also know that A to the 4.5 equals B. And if I know this, I also know that A to the 3.7 equals C. And that means that I can write B over C, which is what's inside the log, as A to the 4.5 divided by A to the 3.7. Now, why is that nice? Are my bases the same, Andrew? What can I do with the exponents, then, if I'm dividing? Which one? You said divide or minus, which one? I can subtract them. In fact, this is A to the 0.8. Is that OK so far? What that really means is I can replace this whole thing as a log base A of A to the 0.8. B over C is the same as A to the 0.8. And now what this is saying here is A to what power equals A to the 0.8 power? Yeah, that's one of those so easy that it's tough question. A to the 0.8 equals A to the 0.8. However, there's a much easier, better way to do that that you just don't know yet. There is a way to do that in your head, showing almost no work. You just don't know the shortcut. Any others? Yeah, I heard 10F. First thing I need to do is get the power by itself. I don't like that 7. What's the 7 doing to the 3 quarters mathematically, Roxanne? So I'm going to move it over. I'm going to rewrite this as 2y over 7 equals 3 over 4 to the x. And now I'm going to write this as a logarithm. Log base of what equals what? Oh yeah, my base is my base is my base. So the base is going to be 3 quarters. The x goes over there. And the 2y over 7 would go inside the logarithm. And I consider that absolutely fair game on your non-calc section of this test, totally. A little tougher, but fair game level difficulty. Is that OK? Turn to the next lesson, lesson 5, page 122. Page 122. And look up. What's the log base 3 of 9? In your head, please. Really? What's the log base 3 of 9 in your head, please? 2. Sharpen your pencil. Next time do it before class starts. OK. What's the log base 3 of 10? That's what we're going to ask ourselves today. A little bigger than 2, I think. Definitely smaller than 3, because if that was a 27, log base 3 of 27 would be 3. And 10 is between 9 and 27. What we're going to ask ourselves is how can we get our calculator to give us an answer? We're going to use something, hurry up. I'm just stalling right now. Thank you. We're going to use something called the base change rule. Is that a costume? Lifeguard? Nice touch. Can you swim? False advertising. If fate has any kind of sense of humor on your way home, you'll walk past the swimming pool, and someone will be crying for help, and they'll be an adult looking at you and beckoning you to come save them. And you'll have to explain to them, I'm sorry. I chose to dress up as a life-saving person, even though I have no life-saving skills, because I thought it was funny. We can only hope. What we're going to look at today are two bases. These are the only two bases your calculator has built in. The first base is what we call the common base or a common logarithm. The common base is base 10. And base 10 is the log button on your calculator, which is right here. So get your calculators out, if you haven't already. And try going log of 10. Find your log button. You have one. Every scientific calculator has a log button. Come in. What's the log of 100? What do you get? By the way, if you're not typing this into your calculator, you're foolish. Not mentioning any names in the back row. There we go. Now I see people typing. You get 2, because 10 squared equals 100. What's the log of 99? Should be a little less than 2. In fact, if you go 2 to the power of 1.995635195, sorry, not 2 to the power of that, 10 to the power of that, because it's base 10, Mr. Dewick. Try that again. If you go 10 to the power of that number there, you get really close to 99. Because actually, this is not the correct answer. Most logs, if they don't work out evenly, they go on and on forever and ever without repeating square roots like pi. So this button here is base 10. And you know how you can remember it's base 10? Look in yellow right above it. There's its inverse, 10 to the power of x. So what's the log of 1,000? I know the answer is 3. And on your calculator, you get 3. What's the log of the cube root of 1,000? Well, there's two ways to do this. I can do this on my calculator log. Where's the cube root button? Look up under your math functions right here. Option number 4 is cube root of 1,000. And then I have to close brackets for the cube root, close brackets for the log, and I get 1. That's one way of doing it. Or Tyson, I could have done this in my head because I know that the cube root of 1,000 is 10. And 10 to what power is 10? 10 to what power is 10? In fact, for years they used to write log base 10. But because it's the common logarithm, if you don't write any base, we assume it's 10 from now on. So if I say what's the log of 8, assume base 10. If I say what's the log base 4 of 8, that's base 4. So all of your calculators have base 10. The second base that they have is base E. Base what? Base E. E is a number like pi. You know how pi is an irrational number? It goes on forever and ever without repeating? E is another number that does that. It's hugely important. This is the first you've seen it. But we'll talk about where it shows up in nature and why it's important in a second. But base E is called the natural log. Natural log because it appears in nature. Natural logarithms are base E, except we don't often write log base E. Instead, we often write ln, which stands for log natural. It's French. And that's on your calculator as well, just below your log button, usually. See the ln button? So on your calculator, what's the ln, the log base E of 5? ln of 5 is? 1.60943. Oh, it said to 1 decimal place. 1.6. What's the log base E of 5? Shannon, it's a trick question. That's 1.6. ln is log base E. We just write it that way. So sometimes they'll write log base E of something. That's ln. What's the ln of E? What's the log base E of E? It's a trick question. E to what power equals E? Right about now, Trevor, you're saying, what is E? I'll show you. On your calculator, if you look right over here, right above the exponent button, there's a yellow pi. See it? That's how you get your pi button. Second function exponent brings up pi. Just below it, what's right above the divided by symbol in yellow? Second function divided by brings up E, and E is, as a decimal, 2.718281828. And it looks like it's repeating, Jessica, because you heard me go 18281828. But right off your screen, it goes all haywire. It's 2.71828, blah, blah, mixture, just like pi, goes on forever without repeating. Those are the only two log bases your calculator has. Log base 10 and log base E. What's so special about this number, Mr. Dewick? Turn the page. First of all, where it says the value of E to 9 decimal places is, can you write 2.718281828, and then a dot, dot, dot, to show that it goes forever without repeating? The number that's on your calculator right now, if you typed it into your calculator. Kirsten, what does every single exponential graph look like? That or that? Now, Kirsten, when we graphed those, we almost always used a base of 2, because we knew our powers of 2, or 3, because we knew our powers of 3. We kept the numbers small. It turns out, when you look at most natural phenomena, population growth, radioactive decay, et cetera, you know what base nature seems to prefer more than any other base? Base E. Base that number, not base 2. Not base, base that number. It likes base E. That's why we call it the natural logarithm, the natural base. If you're graphing, for example, a radioactive decay of a substance, there's various ways to graph it, but the most accurate one will be base E. It also has some wonderful, wonderful properties in calculus, but none of you are in calculus. Are you also a lifeguard? Do you guys connect somehow, or just, can you swim? Eh, at least she's only less of a fraud than you. What's the log base 5 of 25? In your head, two, right? What's the log base 5 of 50? What's the problem with this one right now? Doesn't work out evenly. Yuck. We're now going to find a way to do this on our calculators, but we're gonna start with this one for which we know the answer. So get your calculators out. What we're going to learn is something called the base change law, the base change law. And the base change law is exactly what it sounds like. It lets you change any yucky base into base 10 or base E. We already said the answer to this. What's the log base 5 of 25? Two, on your calculator, what's the log of 25 divided by the log of five? This is base 10, base 10. Try typing that in right now, please. Button 25, close bracket divided by log button five, close bracket. And what do you get? Also two, on your calculator, try going log base E. LN of 25 divided by LN of 25 divided by LN of five. And if you're using a scientific calculator, you should have an LN button as well. It's there somewhere. Also two, this suggests that if I have something in base five, I can write it in base 10 by putting what's inside the log on top of a fraction. And my base in the bottom of the fraction. This suggests that if I want to evaluate log base five of 50, I could go log of 50 divided by log of five. Try typing that in. And then let's see if, right, if we go five to the power of 2.43, I won't type all the rest of the decimal. That should be really close to 50. Is it? Yeah. This is the base change law. It says, Amanda, if they give you a yucky base, put it as a fraction, base 10, base 10. Or base E, base E. What's the log base three of 243? Now I happen to know that this is five. What if I couldn't remember? Well, that's also the same as the log of 243 divided by the log of three. Missed a divided by, Mr. Duk. Let's try that again with a divided by five. Not only is it the same as the log of 243 divided by the log of three, I could have gone ln, natural log of 243 divided by ln of three. Five. Could you give me a favor and just shut that door over there quickly? They're doing some drama stuff in the hallway, I think. Thank you, Tyson. In fact, this, shut the door quickly. In fact, I can write this as log of any base over log of the same base. In other words, don't write this down. If my calculator had a base seven, that would work. Or if my calculator had a base 18, that would work. Or if I really wanted to scare somebody if my calculator had a base pi, that would work. All of those work out to five. This brings us to our base change law, the base change identity. And you need to memorize this. And Steph, here's what it says. If they give you a log and it's a yucky base, you can change it to whatever base you want. By writing it as a fraction, the base that you want is the base in both logs. What's inside the log on the top, you're base on the bottom. This is not on your formula sheet, you need to memorize it. How would I evaluate the log base four of 1,024? I would go the log of 1,024 base 10 divided by the log base 10 of four. What is the log base four of 1,024? What'd you get? You need to pick up your calculators and type in practice. Pick up my hint please. Five? You get the same answer with LN instead of log? Oh, nerd used LN. Yeah, I like that too. Makes me feel like more of a nerd. As long as you use the same top and bottom, you're fine. Man, do you get it okay? Yeah. What if they gave me something like this? Or what if they gave me something like this? The log base 12 of 1,024 over the log base 12 of four. Because that's also the same thing, by the way. These are all interchangeable. Well, what they're really doing is they're giving me the right-hand side. See how I have the same base top and bottom? See how I have the same base top and bottom? Write it as the left-hand side. This is the log base four of 1,024. And how can I evaluate that on my calculator? Log of the top divided by log of the bottom. Although we just did that in A, did we not? So now that we have our base change law, we can evaluate logs of any base. It says evaluate the following logs to the nearest hundred by changing the base. What's the log base five of 221? Well, that's going to be the log of 221 divided by the log of five. 3.35. What's the log base two of 1 over 1,000? It's going to be the log of 1 over 1,000. Close the bracket off, Mr. Dewick. Divided by the log of 2. Oh, heck, just for giggles. Let's use the LM button, Mr. Dewick, because you can't let Andrew out-nerd you. Do you get negative 9.97? C, 3 log base seven of 512. What do you think the 3 is doing to the log mathematically? Okay, so let's do that. 3 times log 512, log 512 divided by log 7. 9.62. 9.62. Not only can I use this to evaluate on my calculator, turn the page. I can use this to write any base as any other base. So example four says, write that as base three. Piece of cake. It's going to be log base three on the top. It's going to be log base three on the bottom. And Jessica, the 216, is going to go inside the top log, and the 6 is going to go inside the bottom log. Mr. Dewick, how would you type that into your calculator? Well, to be honest, I wouldn't type this into my calculator. I'd go if they wanted an answer, log 216 divided by log 6. Sometimes rewriting something as a different base makes life easier. Stuff will cancel, things like that. B, write log of 300 as base five. Okay, it's going to be log base five over log base five. The 300 is going to be inside the top log, and the five is going to be inside the bottom log. Sorry, not a five here, Mr. Dewick. What's my base right here? It's invisible. They said if we don't write a base, what is the base? Sorry? Ten. That's the common base. Not one. In fact, the log base can't be one. That's restricted, because one wouldn't work. It's the same reason one doesn't work with exponents very well. Okay. Example five says, find the exact value of the following. Log of one over 729 divided by log of three. Negative six. See how I did that, okay? Bring that back up, Mr. Dewick, so we can see. On your own, try doing B, C, D, E, and F. Try these on your own. I'm doing them in my head. I'm not getting these wrong. If I'm wrong, let me know. I did all these in my head. Am I wrong? No? By the way, thank you guys for shouting the bell. Is there any of those you want me to go over? Yeah. D? You know what? I would not do D on my calculator. Do the exponent first, because bed mass says E for exponents. Look at the exponent. Roxanne, read the exponent to me. Sorry to say it again. Log base five of 25. What is the log base five of 25? In your head. That whole thing is five squared. It's easier to do in your head than going to calculator there. By the way, if you were doing it on your calculator, Roxanne, you would go like this. Five to the power of bracket. Log 25 divided by log five. And now close the exponent off. I'm going to suggest it's just quicker to do the one you had. In fact, there's an even better way to do that one. There's a shortcut that I haven't taught you yet. And I got negative five and negative 10. Are there any of those ones you want me to do? Yeah. Sorry. E? Log square root one over 1024. Close off the square root bracket. Close off the log bracket. Divided by log two. You got to be careful. Look, your brackets got to be, you know, you got to close off your brackets. They got to be symmetrical. They have to, for every open there has to be a closed bracket. Same idea for F. Log of, and you would go 49 to the power of negative five. Close bracket divided by log seven. Close bracket. And there's your negative 10. Oh, there's a better way to do that one. Is that okay? It's a shorter lesson today. You're going to have lots of time on the homework, partly because it lets you go trick-or-treating. For those of you who are still young enough to go. Number one, all. Two, all. Three, all. In fact, you know what the homework is? Everything is skip six. So four, all. Five, all. Seven, all. Eight, all. Nine, all. Ten, all. It sounds like a lot. It's not bad at all. My kids all finished it during class time.