 This video will talk about properties of logs, so the fundamental properties of logs are given here in this box and that's saying that base is greater than zero and B can't be one. So the first one just is log base B of B equal one. That means the exponent on B that gets you B is one. So B to the first is kind of like if we converted it to B, B to the first is equal to B. And then here we have the exponent on B that will give you one is B to the zero. And then here we have log base B of B to the x is just equal to x. And that's, if I convert it again, I have B to the x is equal to B to the x. So they are the same thing. So we can just say that when we have the bases of the same, the exponents are our solutions, so it's just x. And then here we have B raised to the log same base of x equal to x. And since log base B of x is equal to, that should say, log base B of x like the opposite idea of number three. This one was exponential, this one's logs, this one's dealing with the log, this one's dealing with an exponential. So we want to take these problems and use the properties to solve them. Well I have a log x here and I would like to be able to do it to be just x. So I should be able to say take the base as in property four, use both of these as my exponents, raise the base to each side. It's what we call exponentiating. So I'm going to say ten because that's the base of my log. And then log x is equal to ten to the 1.6. And this is the property. If the bases are the same, then it's just equal to that argument x. The bases are the same up here, so it's equal to this argument x. So here we have x and then here we have ten to the 1.6. And I'm going to be happy with that right now. So over here now we have an exponential and I want to get to just x. So I can use the log property over here, property three, which says my bases can be the same. Well if I have an e, that means I have to take a natural log. So the natural log of base e of my e to the x is going to be equal to the natural log of that 0.343. And when we do that, because this is ln e and the base was e, I can just say that it's my exponent is equal to ln of 0.343. And we'd be done. And we have one more to solve here. So let's get a little bit more involved, but we can keep working. So ten to the 2x is going to be equal to, if I subtract 27, that's going to give me 163. And now I have this exponent. So let's remember those two properties. We had log base b of b to the x was equal to x. So can I take the log? Or we had v to the log base b of x was also equal to x. So if I'm starting with an exponential, that means I want to take the log of both sides and I want to take the log base of my exponent. So I want a log of ten to the 2x is equal to log of 163. My base of my log and my base of my exponent are the same. So the exponent is what I have left. So we have ten versus undo each other. It's like taking a square root and square and they cancel each other out. And then we have log 163 on this side. And if we divide by 2, x is going to be equal to the log of 163. And then we'll divide that whole thing by 2. So find the exact and approximate solutions to these things and we have this problem over here. So we are going to subtract our 175 e to the point zero five x plus one is equal to and that's a thousand when we subtract our 175. And I want the base by itself. I need to get that base completely by itself before I can start using my properties. So I need to divide by 250. Now I have e to the point zero five x plus one is equal to four. And now I'm ready for my property. And again, since I have an exponential, I'm going to do the log same base. So it's a natural log of e to the point zero five x plus one is equal to I took the natural log on that side. So I had to take the natural log on this side, the natural log of four and the natural log and the base cancel each other out. And we're just left with point zero five x plus one is equal to four l in four point zero five x is going to be equal to l in four. That's a number minus one. So don't change this and say it's three because l in four is not four. And then we want to divide it by point five. So x is going to be equal to l in four minus the one and all of that will be divided by point zero five. Another one. So we are trying to get to this. We need to get rid of everything else except for the l in three x. So the first thing you do is start adding and subtracting across the equal sign. So we're going to subtract seven, which will give us negative 22 on this side is equal to negative eight l in three x. And then I have to divide off my eight and you can either reduce it or do whatever you want to with it. We can leave it this way is equal to l in three x. Well, this is going to end up being a positive. Let's go ahead and simplify it. They're both divisible by two. So it would be 11 over four is equal to l in three x. And I want to now I have a log. So I need to exponentiate. I need to get the base of my log, which is E, and then make my log my exponent on that base. So l in of three x that looks like three to the x over here. This is just x and then that's equal to I still have to take E. If I raise the one side to the E, I had to raise the other side to the E and it's 11 over four. So here I have E to the 11 over four. And on this side, we have the base of E and the natural log are going to cancel each other out because they're inverse functions of each other, leaving us with that argument because the property said same basis. We can just have the argument be our solution and then we just need to divide by three. So E to the 11 over four divided by three is going to be equal to our x. We're trying to get the log two x. So first thing I need to do is take the nine to the other side and then I'm going to divide by negative four. So log of two x is going to be equal to negative five point four divided by four. I'll just leave it that way except I will make it since I divided by negative four. I will make it a positive number on this side. And now we are ready to exponentiate. This is base ten. So I'm going to say ten log base ten of two x, which is just log two x is equal to ten to the five point four divided by four. We have this property over here that says, and then we use our property. The property says same base on the base raised to a log with the same base would just be equal to the argument or two x. And over here we have ten to the five point four divided by four. And we just have to divide by two and we'll be done. Ten to the five point four divided by four all over two. There's our final answer.