Hi, I’m Kendall Roberg, and this is Properties of Logarithms. This is the product property. If you have log base b of m times n, you can rewrite this as log base b of m plus log base b of n. Let’s see an example of this. If we have log base 4 of 21, we could rewrite this as log base 4 of 3 plus log base 4 of 7. In this situation, we’re saying 3 times 7 equals 21, so our m is 3 and our n is 7. So we can rewrite it as log base 4 of 3, our m, plus log base 4 of 7, our n. This is the quotient property. If we have log base b of m divided by n, this can be rewritten as log base b of m minus log base b of n. Let’s see an example of this. If we have log base 6 of three-eights, our m is our 3, and our n is our 8. So we can rewrite this as log base 6 of 3 minus log base 6 of 8. This is the power property. If we have log base b of m raised to the n power, we can rewrite this as n times log base b of n. The exponent simply moves to the front of the log. Let’s see an example of this. If we have log base 7 of 3 to the fourth power, we can rewrite this as 4 times log base 7 of 3. But how can we use these properties? Well, let’s suppose we’re given two logarithm values. log base 4 of 3 is approximately equal to 0.792. and log base 4 of 7 is approximately equal to 1.404. Given these two values, we can find tons of other values using the properties of logarithms. Like, let’s try to find log base 4 of three-sevenths.

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Here, we can use the quotient property to rewrite this as log base 4 of 3 minus log base 4 of 7. We know these two values because they’re given above. So now all we have to do is subtract. 0.792 minus 1.404 equals -0.612. So, log base 4 of three-sevenths is -0.612. Let’s try this second one. Here, we’re trying to find log base 4 of 21. Well, 3 times 7 is 21, so if we rewrite this as log base 4 of 3 times 7, we can use the product property. log base 4 of 3 times 7 is really just log base 4 of 3 plus log base 4 of 7. We have these two values above, so now all we have to do is add. 0.792 plus 1.404 equals 2.196. Now, let’s look at the last one. Here, our challenge is to find log base 4 of 49.

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Well, if we rewrite 49 as 7 squared, we can use the power property. The power property allows us to bring the exponent out to the front as a multiplier or coefficient of the log. So now we have 2 times log base 4 of 7. Well, we know what log base 4 of 7 is, so let’s substitute in that value. 2 times 1.404 is just 2.808. Another application of the properties of logarithms is expanding logarithms. Here, we have log base 6 of 5x cubed over y. Because this is a fraction, we can use the quotient property to rewrite this as log base 6 of the numerator minus log base 6 of the denominator. Now over here, we recognize that this is 5 times x cubed, so we can use the product property to rewrite this first part as log base 6 of 5 plus log base 6 of xcubed. Now this just stays the same; there’s not much we could do with log base 6 of y. So we’ll keep it as minus log base 6 of y. Now, here we see x cubed. This means we can use the power property to rewrite this piece as 3 times log base 6 of x. And then we need to keep these ones the same, so log base 6 of 5 plus the one we just rewrote minus log base 6 of y. This process of taking this original log and expanding it is called “Expanding.”