Proof of the logarithm quotient and power rules

let's see if we can stumble our way to a another logarithm property so let's say that oh I don't know let's say that the log base X of a is equal to B right that's the same thing as saying that is the exact same thing as saying that X to the B is equal to a fair enough so what I want to do is experiment what happens if I multiply this this expression by another variable let's call it C right so I'm going to multiply both sides of this equation times C now let's switch colors just to keep things interesting that's not an X that's a ch4 I just to a dot instead times C right so I'm gonna multiply both sides this equation times C so I get C C times log base X of a is equal to multiply both sides the equation right is equal to B times C fair enough I think you realize I have not done anything profound just yet let's go back we said that this is the same thing as this so let's let's experiment with something let's raise this side to the power C so I'm going to raise this side to the power C that's a kind of a caret and when you type exponents that's what you would use a carrot right so I'm going to raise it to the power C so then this side is X to the B to the C power is equal to a to the C right all right it hasn't raised both sides of this equation to the C power and what do we know about what do we know about when you raise something to an exponent and you raise that whole thing to another exponent what happens to the exponents well that's that's just an exponent rule and you just multiply those two exponents so this just becomes this just implies that X to the B C is equal to a C what can we do now well I don't know let's take the logarithm of both sides okay or let's just write this let's not take the logarithm of both sides let's write this as a logarithm expression we know that X to the BC is equal to a to the C well that's the exact same thing as saying that the logarithm base X of a to the C right logarithm base X of a to C is equal to BC correct because it all I did is I rewrote this as a logarithm expression and I think now you realize that something interesting has happened that BC this BC well of course it's the same thing as this BC right so this expression must be equal to this expression and I think we have another logarithm property that if I have some kind of coefficient in front of the logarithm I'm multiplying the logarithm so if I have C klog o'clock base X of a but that's C times the logarithm base X of a that equals the log base X of a to the C so you could take this coefficient instead make it an exponent on on the term inside the logarithm that is another logarithm property so let's review what we know so far about logarithms we know we know that if I write let me let me use a different let me say if I write well let me just letters I've been using C times logarithm base X of a is equal to logarithm base X of a to the C we know that and we know we just learned that logarithm base X of a plus logarithm base X so B is equal to the logarithm base X of a times B now let me ask you a question what happens if instead of a positive sign here we put a negative sign well you could probably figure it out yourself but we could do that same exact proof that we did in the beginning but in this time we will we'll set it up with a negative so if I said that let's just say that log base X of a is equal to L let's say that log base X of B is equal to M and let's say that let's say that log base X of a divided by B is equal to n right now let's how could we write all of these expressions as exponents well this just says that X to the L is equal to a alright let me let me switch colors I keeps it interesting this is just saying that X to the M is equal to B and this is just saying that X to the N is equal to a over B right so what can we do here well what hot was another way of writing a over B well that's that's just the same thing as writing X to the L because that's a X to the L over X to the M that's B right and this we know from our exponent rules right it's just you could write this could also be written as X to the L X to the negative M right or that also equals x to the L minus M so what do we know we know that X to the N is equal to X to the L minus M right X to the N is equal to X to the L minus M those equally gender I just made a big equal chain here so we know that n is equal to L - M what does that do for us well what's another way of writing n I'm going to do it up here because I think we have stumbled upon another logarithm rule what's another way of writing n I did it right here this is another way of writing n right so logarithm logarithm base X of a over B this is an x over here is equal to L L is this right here write log base X of a is equal to L the log base X of a minus M I wrote em right here that's log base X of B log base X of B there you go I probably didn't have to prove it you could apply you try it out with tried it out with you know dividing it putting or whatever but and you now are hopefully satisfied that we have this new logarithm property right there I have one more logarithm property to show you but I don't think I have time to show it in this video so I will do it in the next video I'll see you soon