Proof of the logarithm product rule

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 base x of a is equal to, I don't know, make up a letter, n. What does this mean? Well, 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 is very important to realize that when you evaluate a logarithm expression, like log base x of a, the answer when you evaluate, what you get, is an exponent. This n is really just an exponent. This is equal to this thing. You could've 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, it really an exponent. And we're going to take that notion. And that's where, really, all of the logarithm properties come from. So let me just do -- what I actually want to do is, I want to to stumble upon the logarithm properties by playing around. And then, later on, I'll summarize it and then clean it all up. But I want to show maybe how people originally discovered this stuff. So, let's say that x, let me switch colors. I think that that keeps things interesting. So let's say that x to the l is equal to a. Well, if we write that 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, it's the same thing, I just switched 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 switched letters. So let's just keep going and see what happens. So let's say, let me get another color. So let's say I have x to the n, and you're saying, Sal, where are you going with this. But you'll 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? Well, let's start with with this right here. x to the n is equal to a times b. So, how could 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. 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. But what's x to the l times x to the m? 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, let me take a neutral color. I don't know if I said that verbally correct, but you get the point. When you have 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. That's kind of onerous 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. Oh, I wanted that to be green. x to the l plus n. 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. These exponents must equal each other. So we know that n is equal to l l plus m. What does that do for us? I've kind of just been playing around with logarithms. Am I getting anywhere? I think you'll 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 that 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 base x of a times b. And what does that equal? Well, that equals l. Another way to write l is right up here. It equals log base x of a, plus m. And what's m? m is right here. So 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 base x of a plus the log base x of b. And this, hopefully, 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.