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Properties of Logs

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Today we're going to talk about properties of logarithms. So we're going to go over a few different properties of logarithms, and we'll do an example of each different property.

So let's start by looking at the product property for logarithms. That property says that if I have some log to base b of x times y, that we can write that as log base b of x plus log base b of y. So we're separating the factors of x and y into a sum of two separate logarithms. So let's see a couple of examples of what that would look like. Let's say I have log base 3 of 10. 10 can be split up into two factors, five and two. So I can write that as two separate logarithms with the same base 3. So that will be log base 3 of 5, plus log base 3 of 2.

We can apply the property in the opposite order, or the opposite way. So let's say I have log 8, plus log of 7. Remember, if there's no base, then that means that it's a common log, or a base 10. Using our property, our product property, I can write that as log, as one single log, multiplying the two arguments, eight and 7 to give me 56.

So next let's look at the quotient property for logarithms. This property says that a log of some base, b, of the quotient, x over y is equal to log base b of x minus log base b of y. So we can separate the x and the y into two different logarithms, combined through subtraction.

So let's look at an example using that property. Let's say I have log base 3 of 10. 10, I know, is the same as 50 divided by 5. So I can write that as log base 3 of 50, minus log base 3 of five.

We can use the property going in the other way. If I have the natural log, which is just the log base e of 6 minus the natural log of, 3-- 6 divided by 3 is 2. So using my property, I know that that's going to be equal to the natural log of 2.

Let's look at the power property for logarithms. The power property says that a logarithm of some base, b, of x to the n n, is equal to n times log base b of x, So our exponent becomes a multiplier in front of the logarithm.

So let's look at an example. Let's say I have log base four of 25. Using my property, I know that 25 is the same as 5 to the second power. So I can write that as log base 4 of 5 1/t power And then using my property, I know that this exponent it becomes a multiplier in front of my logarithm. So two times log base four of. Five

We can use the property going the other way out so. Let's say I have 5 times, log base six of three. Using my property, I know that this multiplier can become that exponent of my argument. So this it becomes log base 6 of 3 to the 5th. Power. And simplifying 3 to the 5th power, I know that's 243. So this becomes last log, base 6 of 243.

So now let's look at something called the change of base formula. The change of base formula says that log of some base b of x is equal to log base a of x, divided by log base a of b. And the change of base formula is useful when you want to evaluate a logarithm with a base other than 10 or e. So other than a common logarithm, or a natural logarithm. Our calculators only have buttons for common logarithms, base 10, or natural logarithms, with a base of e.

So if you're using your calculator, if you press the log button, it assumes that you are using the logarithm has a base of 10. So let's look at an example of how we can evaluate a logarithm using the change of base formula for a base other than 10.

Let's say we have log base 6 of 1,296. So we want to evaluate that logarithm, which means we want to find what exponent, 6 to what exponent, will give me 1,296. Again, because our calculator doesn't have a log base 6 button on it, we have to use our change of base formula.

So using our formula, I see that I will have log base A. And my value of A is going to be 10, because I want to use my calculator button, which assumes a base of 10. So I can write log base 10 here, but I also know that when we have a base of 10, we can just not write it. So I'm just going to leave this as regular log. My value for x is 1,296. So I have log 1,296 over log base a. Again I'm just going to use log base 10, so I'm not going to write 10 for my base. And then my value for b is going to be 6. So in the denominator I have log 6.

Now we want to evaluate this in our calculator, log of 1,296 and log of 6 are both decimals. So if we evaluate both of those quantities separately, and then divide them, we may have a less accurate answer due to rounding. So the best way to get your most accurate answer in your calculator is to punch this in using a parentheses-- log 1,296, close your parentheses, divided by log, in parentheses again, 6. And when you punch it in one time like that, you don't need to worry about rounding. And you see that the answer will be four. Which makes sense, because six to the fourth power gives us 1,296.

So finally let's look at a couple other logarithmic properties. The first one is log base b of b is equal to 1. So regardless of their value for b, if the base and the argument of a logarithm is the same value, then it's going to be equal to 1. We Can see why this is by converting this into exponential form. In exponential form we will have-- our base, b, will stay the base. Our exponent becomes the output, and that's equal to our argument, or the input. And we know that this is true, based on what we know what exponents. Anything to the first power is just that same value.

We can also look at a second property for logarithms . And that is log base b of 1 it always going to be equal to zero, no matter what your base is. So again we can see why that's true by converting this into exponential form. In exponential form, this was their base, B. Our exponent it will be zero, and it should equal 1.

And again, thinking about our properties of exponents, we know that's true. Anything to the 0 power is always going to be equal to 1.

So let's go over a few key points from today. The product, quotient, power, change of base and other properties of logarithms are used for simplifying and solving logarithmic expressions and equations. Common log and natural log are usually the only logs that can be calculated directly on the calculator. The change of base formula can be used for calculating logs with other bases.

So I hope that these key points and examples helped you understand a little bit more about logarithms. Keep using your notes, and keep on practicing. And soon you'll be a pro. Thanks for watching.

Formulas to Know

- Change of Base Property of Logs
- Other Properties of Logs
- Power Property of Logs
- Product Property of Logs
- Quotient Property of Logs