Using the logarithmic product rule

We're asked to simplify log base 3 of 27x. And frankly, this is already quite simple. But I'm assuming they want us to use some logarithm properties and manipulate this in some way, maybe to actually make it a little bit more complicated. But let's give our best shot at it. 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? 27x is the same thing as 27 times x. 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. This is equal to the logarithm base b of a plus the logarithm base b of c. 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. So let me make that a little bit clearer to you. And if this part is a little confusing, the important part for this example is that you know how to apply this. But it's even better if you know the intuition. So let's say that log base b of a times c is equal to x. So this thing right over here evaluates to x. Let's say that this thing right over here evaluates to y. So log base b of a is equal to y. And let's say that this thing over here evaluates to z. So log base b of c is equal to z. Now, what we know is, this thing right over here or this thing right over here tells us that b to the x power is equal to a times c. Now, this right over here is telling us that b to the y power is equal to a. And this over here is telling us that b to the z power is equal to c. Let me do that in that same green. So I'm just writing the same truth. I'm writing it as an exponential function or exponential equation, instead of a logarithmic equation. So b to the zth power is equal to c. This is the same statement or the same truth said in a different way. And this is the same truth said in a different way. 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. We know that already-- times b to the z power. 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. And so if b to the y plus z power is the same thing as b to the x power, that tells us that x must be equal to y plus z. If this is confusing to you, don't worry about it too much. The important thing, or at least the first important thing, is that you know how to apply it. And then you can think about this a little bit more, and you can even try it out with some numbers. You just have to realize that logarithms are really just exponents. I know when people first would tell me that, I was like, well, what does that mean? But when you evaluate a logarithm, you're getting an exponent that you would have to raise b to to get to a times c. But let's just apply this property right over here. 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 this right over here, we can evaluate. This tells us, what power do I have to raise 3 to to get to 27? You could view it as this way. 3 to the ? is equal to 27. Well, 3 to the third power is equal to 27. 3 times 3 is 9 times 3 is 27. So this right over here evaluates to 3. So if we were to simplify-- or I guess I wouldn't even call it simplifying it. I would just call it expanding it out or using this property, because we now have two terms where we started off with one term. Actually, if we started with this, I'd say that this is the more simple version of it. But when we rewrite it, this first term becomes 3. And then we're left with plus log base 3 of x. So this is just an alternate way of writing this original statement, log base 3 of 27x. So once again, not clear that this is simpler than this right over here. It's just another way of writing it using logarithm properties.