Using the logarithmic product rule

we're asked to simplify log base 3 of 27 X 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 we have to raise 3 3 2 to get 27 X 27 X is the same thing as 27 times X and so the logarithm property we it seems like they want us to use is log base let me write it log base B of 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 this part is a little confusing the important part for this examples that you know how to apply this but it's even better if you know the intuition so let's say that log 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 we know is this thing right over here this thing right over here or this thing right over here tells us 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 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 as an exponential function or an exponential equation instead of a logarithmic equation so B to the Z power is equal to see these are this is the same statement this is the same statement or they're the same truth set in a different way and this is the same truth set in a different way well if we know that if we know that a is equal to this is equal to B to the Y we can end C and C is equal to B Z then we can write B to the X power is equal to B to the Y power due to the Y power that's what a is we know that already times B to the Z power times B to the Z power and we know from our exponent properties 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 under the 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 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 and I know when people first tell me that I was like what 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 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 question mark 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 guess I would call it simplifying and 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 so 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 27 X so once again not clear that this is simpler than this right over here is just another way of writing it using using logarithm properties