# Write as the sum or difference of logarithms

And this comes straight out of the exponent properties that if you have two exponents, two with the same base, you can add the exponents. Since we are interested in separating the x's, y's and z's into separate terms we will be using the first three properties from left to right.

Let's say that this thing right over here evaluates to y. Well, 3 to the third power is equal to It's just another way of writing it using logarithm properties. And we know from our exponent properties that if we take b to the y times b to the z, this is the same thing as b to the-- I'll do it in a neutral color-- b to the y plus z power.

This comes straight out of our exponent properties. I know when people first would tell me that, I was like, well, what does that mean? You could view it as this way.

## Change of base formula

This comes straight out of our exponent properties. Well, if we know that a is equal to this, is equal to b to the y, and c is equal to bz, then we can write b to the x power is equal to b to the y power-- that's what a is. Used from left to right, this property can be used to "move" of the argument of a logarithm out in front of the logarithm as a coefficient. Let's say that this thing right over here evaluates to y. You can put this solution on YOUR website! The important thing, or at least the first important thing, is that you know how to apply it. And so the logarithm property it seems like they want us to use is log base-- let me write it-- log base b of a times c-- I'll write it this way-- log base b of a times c. So the logarithm property that jumps out at me-- because this right over here-- we're saying what power do we have to raise 3 to to get 27x? So let's say that log base b of a times c is equal to x. We know that already-- times b to the z power. So if we apply it to this one, we know that log base 3 of 27 times x-- I'll write it that way-- is equal to log base 3 of 27 plus log base 3 of x. And then we're left with plus log base 3 of x. So let me make that a little bit clearer to you. It's just another way of writing it using logarithm properties. So if we were to simplify-- or I guess I wouldn't even call it simplifying it.

This is the same statement or the same truth said in a different way. But when we rewrite it, this first term becomes 3.

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