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Hello.

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Let's do some work on logarithm properties.

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So, let's just review real quick what a logarithm even is.

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So if I write, let's say I write log base x of a is

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equal to, I don't know, make up a letter, n.

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What does this mean?

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Well, this just means that x to the n equals a.

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I think we already know that.

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We've learned that in the logarithm video.

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And so it is very important to realize that when you evaluate

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a logarithm expression, like log base x of a, the answer

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when you evaluate, what you get, is an exponent.

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This n is really just an exponent.

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This is equal to this thing.

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You could've written it just like this.

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You could have, because this n is equal to this, you could

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just write x, it's going to get a little messy, to the log

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base x of a, is equal to a.

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All I did is I, took this n and I replaced it with this term.

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And I wanted to write it this way because I want you to

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really get an intuitive understanding of the notion

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that a logarithm, when you evaluate it, it

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really an exponent.

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And we're going to take that notion.

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And that's where, really, all of the logarithm

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properties come from.

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So let me just do -- what I actually want to do is, I

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want to to stumble upon the logarithm properties

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by playing around.

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And then, later on, I'll summarize it and then

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clean it all up.

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But I want to show maybe how people originally

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discovered this stuff.

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So, let's say that x, let me switch colors.

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I think that that keeps things interesting.

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So let's say that x to the l is equal to a.

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Well, if we write that as a logarithm, that same

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relationship as a logarithm, we could write that log base x of

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a is equal to l, right?

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I just rewrote what I wrote on the top line.

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Now, let me switch colors.

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And if I were to say that x to the m is equal to b, it's the

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same thing, I just switched letters.

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But that just means that log base x of b is

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equal to m, right?

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I just did the same thing that I did in this line,

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I just switched letters.

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So let's just keep going and see what happens.

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So let's say, let me get another color.

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So let's say I have x to the n, and you're saying, Sal, where

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are you going with this.

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But you'll see.

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It's pretty neat. x to the n is equal to a times b.

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x to the n is equal to a times b.

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And that's just like saying that log base x

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is equal to a times b.

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So what can we do with all of this?

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Well, let's start with with this right here.

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x to the n is equal to a times b.

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So, how could we rewrite this?

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Well, a is this.

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And b is this, right?

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So let's rewrite that.

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So we know that x to the n is equal to a.

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a is this.

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x to the l.

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x to the l.

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And what's b?

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Times b.

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Well, b is x to the m, right?

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Not doing anything fancy right now.

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But what's x to the l times x to the m?

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Well, we know from the exponents, when you multiply

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two expressions that have the same base and different

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exponents, you just add the exponents.

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So this is equal to, let me take a neutral color.

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I don't know if I said that verbally correct, but

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you get the point.

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When you have the same base and you're multiplying, you can

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just add the exponents.

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That equals x to the, I want to keep switching colors, because

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I think that's useful.

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l, l plus m.

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That's kind of onerous to keep switching colors, but.

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You get what I'm saying.

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So, x to the n is equal to x to the l plus m.

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Let me put the x here.

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Oh, I wanted that to be green.

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x to the l plus n.

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So what do we know?

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We know x to the n is equal to x to the l plus m.

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Right?

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Well, we have the same base.

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These exponents must equal each other.

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So we know that n is equal to l l plus m.

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What does that do for us?

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I've kind of just been playing around with logarithms.

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Am I getting anywhere?

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I think you'll see that I am.

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Well, what's another way of writing n?

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So we said, x to the n is equal to a times b -- oh, I

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actually skipped a step here.

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So that means -- so, going back here, x to the n

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is equal to a times b.

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That means that log base x of a times b is equal to n.

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You knew that.

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I didn't.

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I hope you don't realize I'm not backtracking or anything.

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I just forgot to write that down when I first did it.

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But, anyway.

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So, what's n?

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What's another way of writing n?

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Well, another way of writing n is right here.

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Log base x of a times b.

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So, now we know that if we just substitute n for that, we

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get log base x of a times b.

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And what does that equal?

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Well, that equals l.

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Another way to write l is right up here.

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It equals log base x of a, plus m.

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And what's m?

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m is right here.

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So log base x of b.

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And there we have our first logarithm property.

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The log base x of a times b -- well that just equals the log

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base x of a plus the log base x of b.

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And this, hopefully, proves that to you.

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And if you want the intuition of why this works out it falls

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from the fact that logarithms are nothing but exponents.

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So, with that, I'll leave you with this video.

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And in the next video, I will prove another

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logarithm property.

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I'll see you soon.

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