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>> This is part one of logarithms,
which is an introduction to logs,
and we do the following four things on this video.
This is part one, which is an introduction to logarithms.
So we're going to consider this function.
F of x equals b to the x where b is greater than 0
and b is not equal to 1.
So recall, this is an exponential function,
and we know exponential functions are one to one,
and so they have an inverse.
Alright. So let's rewrite it using our xy notation.
We'll say that f is just the function f in place of f
of x. Remember, I write y equals b to the x. OK.
So that means how do we get f inverse?
We just switch the x and the y's.
So we write x equals b to the y. But then we have
to solve for y. Oh, oh.
Here's the problem.
How do I solve for y when it's in the exponent?
Well, we, we have to come up with a brand-new notation,
and that's what logarithms are all about.
So I'm going to give you definition here for another way
of writing x equals b to the y. So another way
of writing x equals b to the y is y equals log
of x base b. Now, notice how I wrote this and how I read it.
I read this as log of x base b. Some people write it,
read it as log base b of x, but this little number indented
down between the log x is called the base, the little b,
and notice it's the same as the base
in the exponential function.
Alright. So this is a brand-new notation,
and this is how you solve for y.
So when you see something written as x equals b to the y,
what you could do is you take the exponent, which is y,
and you say it's equal to the log of the other number, right,
the log of x, and whatever the base is, you put that down here.
OK. So the first thing we need
to do is basically practice doing that, but before going on,
let's just write now, rewrite that y as f inverse
of x equals a log of x base b. So here we go.
There's some things to notice.
If f of x equals b to the x, then f inverse
of x is the log of x base b. Alright.
So notice for f inverse, y is in the, is the exponent.
So now I've got that the exponent equals the log
of x. That's one thing you want to remember.
Logs are really just exponents.
It'll take awhile to get used to this notation
and to fully understand it,
but eventually it'll make a little bit more sense.
It still is true that b has to be greater than 0
and b is not equal to 1.
Alright. So it's the same value as b, etc. OK.
Let's summarize slightly.
Log is the abbreviation for the word logarithm.
And if b is greater than 0, and b is not equal to 0,
not equal to 1, then this statement right here,
which we read as the log of a base b equals n,
is the same as this equation, b to the n equals a. So I'm going
to write down what this, how you read this.
We read it log of a base b equals n. So here's an equation,
and when you look at it, it looks kind of funny,
but here's how you write it in exponential form.
This is what you do.
You take a look at what the base is.
So that's that little number down here,
and you write b. Then you go
to the other side of the equals sign.
So b to that exponent.
OK. So b to the n equals a. Alright.
Equals that a. So you have to get used to changing back
and forth from one to the other.
So that's the first thing we're going to do.
So here's the definition basically.
The log of a base b equals n is the same thing as b
to the n equals a. We have log form.
We have exponential form.
So for each of these log forms, I've given you three,
I want you to write the equivalent exponential form.
So put the video on pause, and try these three problems.
OK. Ready?
So how we going to do it.
You look down.
The base is 3.
So we write 3 to the x, right.
You put this other number over here.
That's what the exponent is.
The log equals the exponent.
Think about it that.
That's the exponent over there.
Three to the x equals 9.
Now, once you have an exponential equation,
you could use the methods we've used previously
with exponential equations to solve, and hopefully,
you'll recognize that x will be 2
because 3 squared is what equals 9.
Alright. How about this next one?
X cubed equals 125.
Remember, you start with a base, and then the number
on the other side of the equals sign is the exponent.
So it's x cubed equals the 125.
And do you know what that is?
Do you know what number cubed equals 125?
It's going to be 5.
Alright. And the last one.
Again, you write the base.
That's the little number down here,
4 to the negative 1 equals x. So if you're going to solve this,
you would just rewrite 4 to the negative 1 as one-fourth,
and that would be the answer.
All I did was ask you to rewrite each in exponential form, but,
in fact, you could have solved for x
in all three of these problems.
Now we've got to go backwards, which I find harder,
but you should be able to do it either way.
So if you're given exponential form,
you should be able to write it as a log.
[ Pause ]
>> Alright.
So we're going to go the other way around.
We're going to write each exponential equation
as a logarithmic equation.
Now, to be honest, the only time you really go backwards writing
it from exponential to log is
when the variables and the exponents.
So those are the only ones I'm going to give you because that's
when it's useful to do this.
Alright. So how am I going to do it?
Well, here's how I do it.
I write log, and I write log, and I look for the base.
Alright. So the first one, the base is 3.
So that, remember, is the little number here.
OK. And then I know the exponent goes on the other side.
So that means the 81 has to be here,
and the 125 has to be here.
Does that make sense?
And then it's going to equal the exponent.
Now, it doesn't matter if you put equals x,
or you could have written x equals log of 81 base 3.
Now, for this other one, I kind of need a little bit more space.
So I'm going to write it underneath, and I'm going
to write this as m minus 2 equals the log of 125 base 5.
You could also have written that as log
of 125 base 5 is equal to n minus 2.
So that's just getting practice going back and forth.
Most of the time we're going to be writing something that's
in log form as something in exponential form.
Let's see if we can simplify a log.
So somebody wants to know what the log
of 16 base 4 equals, for instance.
Well, remember, what I said that a log was.
It was just the exponent.
So what you're asking is this,
this is the question you're asking yourself.
Four to what power is going to be 16.
So can you figure that out?
Four to what power would be 16,
and hopefully you got that that would be 2.
Make sense because four squared equals 16.
So you could see that would be true.
Now, so you might be able to do it in,
by inspection, but perhaps not.
Maybe that didn't come to you.
So what you could do is say, well,
I was going to let it be a variable, and I'm going to try
to solve for n, and now I take this
and write it in exponential form.
So 4 to the n equals 16.
Now, at this point, you have to write both sides
of the same base if you can, right.
And so you could write both of them with four - that'll work.
So I could write 4 to the n equals 4 squared.
And now since the base is the same, the exponents are equal,
and that's another reason why I now know
that answer is simply 2.
So then you can go back and say, oh, the answer's 2.
You could have also right here, you could have gone
down to the, the base of 2, right.
That would be 2 squared to the n power equals 2 to the fourth,
and then 2 to the n equals 2 to the fourth,
and then made the exponents equal.
You're still going to get n equals 2.
So you could see there's three different ways
that somebody might have gotten that.
The log of 16 base 4 was equal to 2.
By inspection, just saying, well,
I know that it's 4 square that equals 16.
So it's 2.
Or they could have set up a little variable,
said the whole thing was equal to some number n,
and figured out what n was equal to, but you aren't solving
for n. I'm just saying simplify.
So in the end, you don't want to write n equals 2
because there was no n in this particular problem, was there?
Just wants to know what the log of 16 base 4 is,
and it's simply the number 2.
Alright. So we're going to be doing a lot more problems
like this on the next video.
That's just a little sampling.
First, you have to make sure that you understand how
to go back and forth from log, log equations
to exponential equations and get the idea
of what a log equation really means.
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