 This video will talk about exponential functions. Exponential functions have a base that is greater than zero and can't be equal to one and there's all real numbers of x that can go into the domain. This defines the base B exponential function. So I have some examples down here. I have two to the x and over here at this table you can see that if I take two to the negative three, if you remember your exponent rule, that means one over two to the third or one eighth and then if I get down here to two to the zero, it's going to be one and so on. So then this graph comes here. So if we look at this graph here, we can see that it is increasing and then we want to talk about what the domains are. So it's increasing the domain. This thing is going like this forever and it's going like this forever. So this is going up and out to the right and it's going down and to the left. So it would be all reals. If we talk about the range, we want to know how high and how low. Well, this graph doesn't really kind of looks like it, but it never really crosses the x-axis. If I take two to the negative sum big number, I'm still going to have a fraction that's not zero. So we have to say that it's going to be from in a parentheses zero, but it goes up to infinity. So now let's look at and see if we can graph g of x is equal to one half to the x where B is between zero and one down here. So if we put negative three, that's going to be one half, but we have to flip it over to take care of the negative. And then we have two to the third, which is going to be eight. So negative three will give us positive eight somewhere way in the sky. And then this should be a negative two here. If I put negative two in there, I'm going to get four. And if I put negative one in there, I'll get two. And if I put zero in there, I'll get one and one half for one, one fourth for two and so on. You can see that this graph is going to be same basic shape, but it's going the other direction because it is decreasing. Now we want to look at the domain again, still going left and right forever. So that's going to be all reals. The range again is going to be from zero to infinity because it's above the x axis. So now we want to graph one ourselves. Now if we do that, let's see if I can make my graph a little bit better. We know, let's talk about the transformations here. So we have x minus three. So this would be three to the x normally. Look very similar to the first one we just looked at, a little bit steeper. But when we take the x and end the function, which is exponential function. So up in the exponent we're subtracting three. That means we're going to go right three. And then this plus two is going to move it up and down. That's what that constant does. So it's going to go up two. Let's see if we can draw in some axes here. It's not really easy to do with my tablet. But if I get in there's, we'll call that our y axis and we'll call this our x axis. And I want to do three to the x first. If I did three to the x, it would three to the zero would be one. Three to the first would be three and so on. So it would be a graph that looked like this. Oops, I missed, but you get the idea. Now we want to look at, take like this point here. We'll go over three, one, two, three, and then up two, one, two. And if we put zero in here, just so that we know what happens on the y axis, when x is zero, we're going to have three to the negative three, which is one ninth. And then that which would be way down here, then go up two. So one, two. So it would be somewhere around here. So this graph, it would look something like this. Domain is still all reals. The range is now tending toward, I probably should say, the range is now tending toward. I probably shouldn't have been quite so low. It's tending toward now this two. This is the better graph. Get rid of this. So the range should be that y is greater than two. So two to infinity. So we can't quite get to two. So the horizontal asymptote would be y equal to, and b is greater than one. It's an increasing function so that we know b is greater than one. All right, so let's talk about the natural number e. And I want you to realize that this is a number. It's not a variable. It's a number. For, it says x equal, it's greater than zero. Then as x goes to infinity, it gets really large. Then this little thing looking here, one plus one over x to the x, will become e. So if you were to put in your calculator one plus one over one million to the one million, you're going to find out that you get close to this number, which is 2.71, blah, blah, blah, blah, blah. And that's with the value of e. And it's a naturally occurring number in science, especially, and in nature. So it's so common, we call it the natural number. So y equal e to the x will be our exponential function using e. So it says find e to the x for f of 10. And really we're just saying, what is e to the 10? And I need to call my calculator up in a minute. And then this one's really saying e to the negative 3. So I've called my calculator up here. And we want to know e. To do find e, you do second ln. And it gives you the e with the caret. And we put in 10, and we're going to get this number, 22,026.46579. I'll put that in in a minute. And then we're going to do again over for f of negative 3, second ln, that gives me e. And then I put in negative 3, and we get .049. So it's a smaller number, because it's a negative. We're saying 1 over e to the third. So when I actually did these, this gave me, oh, thanks for going back again. The 10 was equal to 22,026, 22,026.47. And e to the negative 3 was .049. So we rounded to .05. So now we want to use our exponentials to solve some equations. And there's a couple of facts that we need to know. If B of m is equal to B of n, the bases are the same. Then we know that their exponents are the same. Equal bases implies equal exponents. If they're not equal, then the bases with the exponents aren't going to be equal to each other either. So we want to solve these problems. So we want to get the same base is what it's really saying. So 12, I can get 12 to the x. Because most of us know that 144 is 12 squared. So if my bases are the same, that means that my exponents are the same. So over here, I'm looking at 9 to the x minus 1. But 27, I can't think of, I mean, 9 squared is going to be 81. So my base can't be 9. But 3 squared would be 9. So I'm going to rewrite 9 as 3 squared. And then it's still raised to the x minus 1. And I know that 3 cubed is 27. So if I do my exponent property that says I can multiply my exponents when one's on the inside, one's on the outside, I'm going to have 3 to the 2x minus 2 is equal to 3 to the 3. Or we can say 2x minus 2 is equal to 3. 2x is equal to x is equal to 5 over 2. And we could double check that with our calculator. We could clear all this out and say, go back and say 9 carats. And then what did we say x was? I've got more than one thing happening in my exponents. So I need a parentheses. And it's 5 divided by 2. And then from that, I'm going to subtract 1. An order of operation says it'll divide before I subtract. And that should be equal to 27. And it is. So we know we did it right. All right. So then we need to do this one. And 1 half. And this is a fraction. This one's not. But I could make 1 half become 2. And that might be a little bit easier to work with. So I'm going to say that this is 2 to the negative first. That would be 1 half is equal to 3x. And I happen to know that 8 is 2 to the third. But I could always check in my calculator 2 to the x and find it where I find it 8. And I'd have x minus 2 to multiply these exponents. I know these exponents are going to be equal to each other. So I'm not going to carry down the base anymore. I'm just going to distribute my exponent. So negative 3x is equal to 3x, distributing minus 6. If I subtract this 3x over here, I'll have negative 6x is equal to negative 6, which tells me then that x is equal to 1.