# Proof of the logarithm productÂ rule

## 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 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.