 Today, we're going to talk about the different properties of logarithms. So on your worksheet, I've posted the different properties of logarithms, and I have them listed here as well. So you should be able to see that they are very similar to our properties of exponents. So the top properties have to do with just whether something is raised to the zero power or to the first power. So since logs are the inverses of exponents, these are kind of the opposite. So property one says a log base b of one is always zero, and then log base b of the number b is one. Then the second and third property will use a lot while we're solving different logarithmic equations. So if you are multiplying two things inside of a log, you can separate it into two different logarithms by adding. And then if you're dividing inside of a logarithm, you can separate using subtraction. But notice the base has to stay the same for each of those. And then our fourth property is the power property. And this just says that if you have an exponent inside of a logarithm, as long as everything inside of that logarithm is raised to the exponent, then you can drop that down in front of the logarithm. Just like we did here with the R and change it to multiplication. So we're just going to look at a couple examples today of how to use those different properties. Okay, first we'll look at a couple examples of how to separate logarithms into more than one logarithm. So the goal is to write this logarithm here as many logarithms as we need to, but we don't want any multiplication, division, or exponents inside of the logarithm. So right here, we have m squared times n. So if you look at your properties to separate multiplication, we can use addition. So this becomes log of m squared plus log of n. Then the last thing that we want to do is anytime we have an exponent, because all of this, the full m, is raised to the second power, this two can go in front and become multiplication. So this just becomes the expression to log m plus log n. Now if we look at the second example, this example here has division instead of multiplication. And we're first going to do the division and then look at the exponents. We can't do them at the beginning because they're opposite, the whole expression isn't raised to the exponent. So first when we have division, we separate using subtraction. And notice I'm keeping the same base, so the natural log for each of these. And then after it's separated, now the full m is to the second power. So I can put that in front of that expression and the three in front of that expression. And so that gives me two natural log of m minus three times the natural log of n. And now we have two expressions with no multiplication, division, or exponents. Let's look at one more example. This example is a little bit trickier than the others because it has a root, a cube root in this case. And a lot of people look at this and don't see anything that we can do to simplify because of that cube root. But I just want to remind you that the cube root is the same as raising something to the one third power. So we're going to start from the outside and work our way in. All of this is raised to the one third power. So right away we can drop that one third in front. So we get one third times the log of three minus v all over two v. And then what I see here is I see this division. And remember division can get separated using subtraction. So I'll change this expression to one third and that one third has to apply to everything. So I'm going to just keep it in parenthesis or I'm just going to do brackets here for now. Three minus v minus the log of two v. Okay so I just got rid of that division. Now if you look here we have three minus v in one logarithm and we can't separate subtraction when it's inside. We can go the opposite way but we can't separate that subtraction at all. But in the second one the two v that is multiplication and we can separate that. So what we would get, we'll keep the one third in front. We're not going to change the first expression. And then this subtraction sign it has to be distributed to this entire log of two v. So we'll say minus log of two plus log of v. Because it is multiplication but we do have this subtraction. So if I get rid of all of the parenthesis just to make it a little bit easier to write we'll distribute the subtraction sign and then we'll also distribute the one third. So I'll just write it in here at the bottom. One third times the log of three minus v minus one third times the log of two minus one third times the log of v. And that gets rid of all parentheses. When we're solving logarithmic equations we're actually going to use the opposite method of the properties where instead of separating them into many logarithms we'll try to combine them into one. So in this case you'll notice what we have is an addition sign. And so if we want to combine these two logarithms into one with addition that means we'll multiply. So we're not going to change the x plus three at all instead we're going to write this just as one logarithm. It's going to be the natural log of x plus three times two x. And all of that is inside of the logarithm. And when we're solving these you'll have to distribute. So we'll just distribute here. We'll distribute the two x. So we get the natural log of two x squared plus six x. And we'll do the same thing below. Here we have subtraction. And subtraction if you look at your properties can be combined into one using division. So since both of these are natural logs we'll just divide. X plus three divided by x minus one. And usually for the division you don't have to do any more work than that. You don't have to do long division in this case. Now we just have one final formula that we will use a lot while evaluating logarithms and it's the change of base formula. So what the change of base formula does is it allows us to evaluate logarithms of any base. So far we only can do logarithms of base ten and base e using those buttons on our calculator. So what it says is if you have a logarithm of any base as long as a is not one and it's a positive real number then we can do the log of the inside of the logarithm. The x divided by the logarithm of the base the a using any base we want. So that allows us to type it into our calculator using just the common log or the natural log. So for instance down at the bottom log base three of sixty seven we could type that into our calculator as log sixty seven divided by log three or natural log of sixty seven divided by the natural log of three either way that will evaluate the expression for us and give us the same answer. Go ahead and try it on your calculator but you should see that whether you do the natural log or the common log you'll get the same answer. I got about three point eight two seven for the solution.