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# Proof of the logarithm product rule

CCSS.Math:

## Video transcript

hello let's do some work on logarithm properties so let's just review real quick what a logarithm even is so if I write let's say I write log log base X of a is equal to I don't know make up a letter and what does this mean well this just means that this just means that X to the N equals a I think we already know that we've learned that in the logarithm video and so it's very important to realize that when you evaluate a logarithm expression like log base X of a the answer or when you evaluate what you get is an exponent right this n is really just an exponent this is equal to this thing right you could have written it just like this you could have because this n is equal to this you could just write X it's going to get a little messy to the log base X of a is equal to a all I did is I took this N and I replaced it with this term and I wanted to write it this way because I want you to really get an intuitive understanding of the notion that a logarithm when you evaluate it really is an exponent and we're going to take that notion and and that's where really all of the logarithm properties come from so let's let me just do what we'll actually want to do is I want to to StumbleUpon the logarithm properties by playing around and then later on I'll summarize them and clean it all up but I want it I wanted to show you maybe how people originally discovered this stuff so let's say that I don't know let's say that X let me switch colors I think that that keeps things interesting so let's say that X to the I don't know L is equal to a well if we write that as a as a logarithm that same relationship as a logarithm we could write that log base X of a is equal to L right I just rewrote what I wrote on the top line now let me switch colors and if I were to say that X to the M is equal to B and same thing I just switch letters but that just means that log base X of B is equal to M right I just did the same thing that I did in this line I just switch letters so let's just keep going and see what happens so let's say let me get another color I might have to run out of kkona I have infinite color and I will never run out so let's say I have X to the N you're revising Sal where are you going with this but you will see it's pretty neat X to the N is equal to a times B X to the N is equal to a times B and that's just like saying that log base X is equal to a times B so what can we do with all of this let's start with with this right here X to the N is equal to a times B so how can we rewrite this well a is this and B is this right so let's rewrite that so we know that X to the N is equal to a a is this is X to the L X to the L and what's B times B well B is X to the M right not doing anything fancy right now well what's X to the L times X to the N well we know from the the exponents when you multiply when you multiply two expressions that have the same base and different exponents you just add the exponents so this is equal to let me take a neutral color and if I said that verbally correct by you get the point when you the same base and you're multiplying you can just add the exponents that equals x to the I want to keep switching colors because I think that's useful L L plus M it's kind of ownerís to keep switching colors but you get what I'm saying so X to the N is equal to X to the L plus M let me put the X here and oh I wanted that to be green X to the L plus M so what do we know we know X to the N is equal to X to the L plus M right well we have the same base we have these exponents must equal each other right so we know that n n is equal to L L plus M okay what does that do for us that's I'm kind of just been playing around with logarithms am I getting anywhere I think you will see that I am well what's another way of writing n so we said X to the N is equal to a times B oh I actually skipped a step here so that means so going back here X to the N is equal to a times B that means that log base X of a times B is equal to n you knew that I didn't I hope you don't realize I'm not backtracking or anything I just forgot to write this down when I first did it but anyway so what's n what's another way of writing N well another way of writing n is right here log base X of a times B so now we know that if we just substitute n for that we get log log base X of a times B and what is that equal well that equals L another way to write L is right up here it equals equals log base X of a plus M and what's M M is right here log base X of B and there we have our first logarithm property the log base X of a times B well that just equals the log of base X of a plus the log base X of B and this is hopefully this proves that to you and if you want the intuition of why this works out it its Falls from the fact that bloggers rhythms are nothing but exponents so with that I'll leave you in this video in the next video I will prove another logarithm property I'll see you soon