>> Now, let's sort of take an expanded version and contract. So let's bring these two together. We have the difference of logarithms, and so that's the logarithm of a quotient. So it's the log of the first item over the second item, as items within that fraction. So the logarithm of this as a numerator and that as the denominator using our quotient rule. All right, when we see coefficients involved in an expression like this, go through and use the power property and write these coefficients as exponents like this. This becomes LN 8 squared then plus LN Z to the fifth. You see, I'm just moving these coefficients to a position of an exponent, you see. And now we can, after doing that, we can bring these together either, often either using the quotient property or the, or the product property. All right, here we have the sum of two logarithms, and so we're going to use the product property. So we have the natural log of the product of 8 squared times Z to the fifth. Now 8 squared is 64, so this is 64Z to the fifth. Let's condense here. Now the first order of business once again is to take the coefficients and move them to a position of, of exponent. So we would have LN 3 to the fourth power. Now on this one, it, it turns out that it doesn't make any difference whether I think of moving negative 2 to that position and putting a plus sign here, or leave the plus sign here and just move the two up there, you see. But what you don't want to do is put negative 2 up there and leave that as a negative. Then you're changing the problem, all right. But either, so either of those techniques works. I kind of like the idea lf just leaving the minus sign here and put the 2 up there.
[ Writing on board ] So we'd have this. All right, now we begin to bring things together. There are a couple of ways of handling this situation with a minus sign. Now, I, I'm going to show you one of them that is real safe.
[ Writing on board ] Let me, let me write 3 to the fourth as 81. That'll make things a little easier. All right. I'm going to factor negative one out of these two, and look at what it does for me. I have LN X squared plus LN Y, and in parentheses, I can use the product property to, to write this as LN X squared Y, you see. And now, because I have the difference of two logarithms, that's the logarithm of a quotient, so it's LN 81 over X squared Y. Now let me show you a faster way to, to look at this.