Hello! Let’s do some work on logarithm properties. So let’s just review real quickly what a logarithm even is. If I write log base x of A is equal to N, what does this mean? Well, this just means that x to the N equals A. I think we already know that, we’d learned that in the logarithm video. 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. This N is really just an exponent; this is equal to this thing. You could have written just like this because this N is equal to this. You could just write x to the log base x of A is equal to A. All I did is I took this N and I replaced 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, it really is an exponent. And we’re going to take that notion and that’s where really all of the logarithm properties come from.
What I actually want to do is I want to stumble upon the logarithm properties by playing around it then later on I’ll summarize them and clean them all up. But I want to just show maybe how people originally discovered this stuff. So let’s say that x to the L is equal to A. Well if we write that as a logarithm, that’s same relationship as a logarithm, we could write that log base x of A is equal to L. If I were to say that x to the M is equal to B and the same thing, I just switched letters but that just means that log base x of B is equal to M. I just did the same thing that I did in this line, I just switched letters. So let’s just keep going and see what happens.
Let’s say I have x to the N is equal to A x B and that’s just like saying that log base x is equal to A x B. So what can we do with all of this? Well let’s start with this right here, x to the N is equal to A x B. So how can we rewrite this? Well A is this and B is this, so lt’s rewrite that. So we know that x to the N is equal to A, A is this, x to the L. And what’s B, times B, well B is x to the N. What x to the L times to the L? Well we know from the exponents when you multiply two expressions that have the same base and different exponents you just add the exponents. So this is equal to, when you have the same base and you’re multiplying, you can just add the exponents, that equals x to the L+M. So x to the N is equal to x to the L+M. Well we have the same base these exponents must equal each other. So we know that N is equal to L+M, and what does that do for us? What’s another way of writing N? So we said x tot eh N is equal to A x B that means that log base x of A x B is equal to N.
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 x B. Now we know that if we just substitute N for that we get log base x of A x B and what is that equal? That equals L, and another way to write L is right up here. It equals log base x of A+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 x 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 falls from the fact that logarithms are nothing but exponents.
So with that, I’ll leave you with this video and in the next video I will prove another logarithm property. I’ll see you soon.