All right everyone, and let's talk about logarithms, everybody's favorite. So even folks have seen these before, let me ask you a question. What is a logarithm? Most people when I ask them this question, even if they've seen logarithms and are comfortable with the rules and all that. They are still loss to kind of describe what it is, so logarhythms you can think of them as two ways. One is that they are specifically the inverse of the exponential function. So let's talk about this for a second. So if I have a number, a, that's not zero, and it can't be one also. So let's get rid of those two cases, and I have a function f(x) = a to the x, a nice exponential function, then it's either decreasing or increasing. We've seen this before, you have exponential growth or exponential decay, so either increasing or decreasing. In particular, right, the graph is one to one. Therefore, it's one to one by the horizontal line test. If I'm one to one, I have an inverse function, therefore there is an inverse function to my exponential and this is what we call a logarithm. So this is what we call, they're called logarithmic functions. Logarithmic, A-R, log L-A, L-O-G-A-R-I, I can't spell logarithmic functions. We'll edit that out in post, no we won't. Okay, called logarithmic functions with a base a. So you can have lots of bases, the one that's going to come up with most, of course, is the natural log. And we denote these functions, if we call it f(x) or f inverse of x, depends on what perspective you're taking. You put the little base down here, the subscript and there's x. Don't get hung up on the fact that the base is called the letter a, sometimes they put b in there for base, if you want. x is also a dummy variable, that could be theta or anything else you want. But these are logarhythms, they're inverse of, they're inverses of exponential functions and therefore they have a nice little rule here. So if I have the logarithm of a(x) is some value, y, that is only true if and only if the base to the output of the logarithm is the input, a(x) = y. We'll do more of these with specific numbers and you can actually see how they interact with each other. Okay, so let's move on. There's our definition, as always, if you get, I find most of the time when you see a definition, it's so abstract that it's a little confusing. Put it down, take note of it, and then always go to specific examples and then play around this thing. So let's jump right to that. So as an example, let's just pick a specific one, log base 2 of x. Okay, so I think about this, this is the inverse, inverse to f(x) = 2 to the x. So whatever 2 to the x does to the function log base 2 will undo it, undo it. So 2 to the x, let's see if we can do this without having to rely on the exponential function. 2 to the x is exponential growth, goes from low to high, and everybody's happy. We know that the graphs of these things are reflections about the line y equals x, so let's think for a second what's going on here. This graph would have to be sort of like this. If it's going to be a mirror image, if I tilt my head 45 degrees of, this way, so what are some special points here, what's going on here? Where is this thing equal to 0? So let's play around this for a second, log 2 of x = 0, where is that true? That's true where 2, so you take the base, raised to the 0, would equal x. So I can tell you exactly what the x value is and where this thing crosses the x to the 0. We all know that, that is 1. So this crosses the x axis, it's x intercept is exactly 1. That's true for actually any log base, so we can put this sort of as a general rule. So if I have log base b as a function that inputs (1), oops, (1), that's always going to be 0. And the reason is because the base to the zero power is always 1. So that's nice, it's a graph. Let's talk about it for a minute in terms of the standard things we ask about graph, what's its domain? So its domain is, it's not all real, be careful, you can't plug in negatives here. If you look at the x axis, the x values you get 0 to Infinity. I am using a parenthesis Sees around 0 because you cannot include 0. What is its range? Its range is a set of y values, a set of Outputs. It goes down to infinity forever, goes up to Infinity forever, so its range is all reals. We have its graph, and the asymptotes, we have a vertical asymptote. A vertical asymptote is like the y axis here. This thing will get close, but it will never touch, and that's at the line x = 0. Has no other asymptotes there. So describe this graph, it goes off to infinity, has nice end behavior, things to know about logarithm. The fact that we use 2, you could pick any number greater than 1, this is just a specific one that we picked. Next, moving on. Log rules, so let's try to put this in a different setting. You've probably seen these in some way before, so let's do log rules. So let's let's put down the one you probably seen. So if I do log to an a of a product is log a of x + log a of y. So logarithms turn multiplication into addition. There's a bunch more of these, we'll write them down in a second, but let's just talk about this for a second. Here's the other way to think about logarithms. Logarithms are an exponent. When you put in exponential function, the variable moves upstairs. So if you think about logarithms as living upstairs in the exponent, so much so that I don't even write the base, we just know we're upstairs in the attic, then this would make more sense. You say, well, why does that make more sense? I don't really understand. If I have, so let's switch over the world of non logs for a second, if I have a base, so like a to the x + y, and I ask you to do some algebra with this, what do you do with this? This becomes a to the x times a to the y. So when I live upstairs, which means logarithms, products become addition. It's a little weird to see, but that's exactly what's happening. You're working upstairs, same thing with division. So let's do log base a x over y. This is log base a of X- log base a of y. When you're working with logarithms, division becomes subtraction. Is that normal? Why, why would that happen? Well, if you take the other side of the coin, and look at division of exponential functions, how do you algebraically simplify this? This becomes 1 base x- y. Division becomes subtraction in the exponent. A logarithm is an exponent. You're working upstairs. One more thing you've seen all the time with logarithms is that they say exponents fall out in front. So let's do something like this. If I have a log to some base a x to the r, then the r falls out in front and it's r times the logarithm. So you'll say, why would that be true? Well, if I take a base, and I have lots of them, so let's say, do this, many, many times, a to the x. What's going on over here? Well, this is a, x there's r of these things, let's say. So it's r to the power, this is a to the x to the r, you're multiplying these together. So I have a power repeated, repeated addition, repeated multiplication. They get multiplied together, and you get a to the xr. In the exponent, raising a power to a power becomes a product, and that's exactly what's happening here. Raising something to a power, maybe you don't even need this, makes more sense. a to the xr raised to a power is a product. That's exactly what's happening in rule number three. This is not really a rule, but I'm going to put it in here anyway, I'll put a little star next to it. Not a rule, but it might as well be. This is something that comes up so much, you might as well just treat it as such. If I plug in 1, you will always get 0 no matter what your base is. We saw this before from the graph. This comes from the fact that if I have an exponential function and I raise it to the 0 power, I'll always get 1. And this is just a common mistake that people make. If you plug in 0. 0, think of the graph for a minute, so what is the log of 0 for any base? This is undefined. Some people will get confused and be like it's 1, and I'm like no, so don't tell me that, it's undefined. Okay, so I put these little two down here, this is a simple thing to review. To play around with logarithms, let's do a quick example just to get some experience. So if I ask you, without a calculator, without running over to the thing, let's do log base 5 of 125. So log base 5 is a function. I say what is this thing? And if you don't like logs, most people who don't like logs, they're pretty good at exponents. You say, well, I'm not going to deal with it. I will take your logs and I'll convert it to an exponent problem. So think of this as what is this thing equal to here? Solve for x, well, I don't like logs, logs are confusing, they don't make any sense to me. So instead, let's rewrite this as 5 to the what = 125. You can convert any log to an exponential problem and vice versa. You can convert any exponential problem to a logarithm. Now it's a little more easier, so 5 squared is 25, 5 cubed, okay, so x is 3, done. Not too bad, right. Just as another example, let's just play around with these things. If I said to you, can we simplify, so log base 2 to the 6- log base 2 to the 15 + log base 2 to the 20. What's going on here? So simplify, write this using a single logarithm. So how do you clean this up a little bit? Just, you don't have evaluate this, or maybe we can, but the first 2 terms notice they all have the same base. One of the rules here is they have to have the same base. If they had different bases I'd kind of be stuck, but they have the same base so I can combine the first two together. Subtraction turns into division so I can write this as a single logarithm log base 2 of 6 over 15. Simplifies a little, we'll clean that up in a second, let's do it now, why not. So log base 2 divided 3, both of them you get two-fifths. I forgot the plus, I'll put that over here, + log base 2 of 20. And now I have two logarithms with the same base, where we're adding. So we go up here to rule number one and I can write that as a product. So this becomes a single logarithm, log base 2(2/5 * 20), and that of course, the 5 and the 20 become a 4, and you get log base 2 of 8. And you say am I done? Well kind of, kind of, you're not wrong if you leave it like this, but 8, remember, is log base 2 of 2 cubed. I'm running out of room here. And when I have powers I can use rule number three to bring the exponent down. So this becomes 3 times log base 2 of 2. Now, here's a question for you, I'm going to put it here as our little things to remember under rule number 4. If I have a log to certain base, and I plug in that base, what is the answer? So let's go over here, let's think of this as an exponential problem. So it's saying a to the what is a? What is the exponent that gives me back the base? Whenever they match, you always get 1, so log 2 of 2 is just 1. This becomes 3 times 1, so our final answer here is 3, okay. All right, review these and then we'll talk more about our natural logarithm, which is the one that's going to come up the most in the class. But keep these handy if you haven't seen this before, or if you need a refresher. I treat number four basically as a rule, it's got three parts. We will use these constantly in this class so get comfortable with that. Okay, we'll pause here and work on these for a bit, before moving on. Talk to you soon.