This section is a review of

the properties of exponents, and then
logarithms.

As I've mentioned before, there's 
not a lot of algebra

in this course, but exponents and logs

are going to come up all the time,

so it's important that you're pretty
comfortable with them

If you're not sure if you need this review

what I'll suggest is the following:

take the quizzes that have interspersed
throughout these videos

and if you can do the quizzes 
without any problem

then there's no need to watch the video.

But if the quizzes are a little bit confusing,

Or you are uncertain about a topic

Then go back and watch the videos that

came right before that quiz.

Let's get started, and we'll begin by

thinking about exponents.

We'll start by thinking about properties

of exponents

The starting point is to think about

what exponents mean, anyway.

So an exponent, as you probably know

means successive multiplication.

For example, 3 raised to the 4th power

means we take 3 and multiply it
by itself 4 times. So

3 to the 4th is 3 times 3 times 3 times 3

which in this case would be... let's see...

3 x 3 is 9, 3 x 3 is 9, 9 x 9 is 81

As long as you remember the definition
of an exponent

as successive multiplication, everything
else follows pretty quickly

What if we had something like

3 to the 4 times 3 to the 2?

Well... 3 to the 4... what's that? 
That's 3 times 3 times 3 times 3

... that's what exponent means.
Multiply by itself 4 times.

3 squared means multiply by itself
2 times... 1... 2...

So we can see... we can just count.
1, 2, 3, 4, 5, 6

So this is 3 to the 6th.

So we see, 3 to the 4 times 3 to the 2
is equal to 3 to the (4 plus 2)

which is how I got 3 to the 6th

You multiply 3 by itself 4 times,
and then 3 by itself 2 times

and multiply those together, that's

the same as multiplying 3 by itself
6 times.

So that's our first property or rule for 
exponents:

x to the a times x to the b is 
x to the (a plus b).

That's important,

so I'll put that in a purple box.

What if, instead of having 3 to the 4
times 3 to the 2,

we had 3 to the 4 divided by 3 to the 2?

So, in other words, what if we had this:

3 to the 4 over 3 to the 2?

Well, 3 to the 4 is 3 times 3 times 3
times 3

3 to the 2 is 3 times 3

so we can cancel...

and we're left with 3 times 3

which is 3 squared.

and so what happened is we had 4 threes
up here

2 threes down here. The downstairs threes 
cancelled two of the upstairs threes

leaving us with 2

so in other words, 3 to the 4 over 
3 to the 2

is 3 to the (4 - 2) which is how we got
3 to the 2.

and that gives us another general property:

x to the a over x to the b is 
x to the (a-b)

And it follows from this that

1 over x to the b is the same thing as 
x to the minus b

So that's another important property.

This analysis tells us what negative 
exponents should mean.

There are a number of ways to see that

Maybe one way is as follows:

Suppose we have 3 to the 5 over 3 to the 7

[Comment] Why did I choose such big numbers?

1, 2, 3, 4, 5... 1, 2, 3, 4, 5, 6, 7.

So, if I cancel, I'm left with

1 over 3 to the 2.

But over here if we use this rule,

This would be 3 to the (5 minus 7)

which is 3 to the minus 2.

So, there are a number of different ways

to convince yourself that this makes
sense.

The bottom line is that a number raised to
a negative exponent

is the same as 1 over a number raised to
that exponent.

So, we have these two general rules for 
exponents.

and these follow directly from the definition
of exponentiation

as successive multiplication.

Let's consider two important special cases

First, what might 3 to the 1st power equal?

A couple of ways to think about this...

Suppose i did this: 3 to the 3 over
3 to the 2?

That's 3 times 3 times 3 over
3 times 3.

And that's just 3. Right?

Those cancel, those cancel and I'm left
with 3.

I could also use this rule. This is 3 to 
the (3 minus 2) which is 3 to the 1.

So 3 to the 1 is just 3.

And that kind of makes sense, right?

3 multiplied by itself one times 
seems like you'd be left with 3.

That's a general thing. That any non-zero 
number raised to the first power

just gives you that number back.

One more to think about...

And that's: what about 3 to the 0?

What should that equal?

Well, let's do something similar to what
I did here...

I'm going to do say 3 squared 
over 3 squared.

I know, I don't even need to 
multiply this out

that's 1... because I have the same thing

at the top and the bottom.

But I could also write this using this...

This is 3 to the (2 minus 2)... 2 minus 2 
is zero

So ha! 
This shows us that 3 to the 0 is 1.

And this is one that sometimes trips
people up.

There's a sense that 3 multiplied by itself

zero times... what should that be?

Kind of feels like it should be zero...

and there's a case to be made for it.

But a number raised to the zero power

is defined to be 1, and the reason is

that then it preserves these nice rules.

The bottom line is x to the 0 equals 1.

That's true as long as x is not equal 0.

Last but not least, let's think about
square roots and exponents.

Suppose I have something like the square
root of 3

I want to know, how can I think about that?

And can I write that as an exponent?

So let's think about what the square root
of 3 means.

By definition, the square root of 3 is a

number which, if I multiply it by itself

square root of 3 times square root of 3

I get 3 back.

We saw just a moment ago that 3 is really

3 to the 1, so I claim that square root 3

has to be 3 to the one-half.

Why? Because the exponent rule says that

I would add these.

3 to the (one half plus one half)

equals 3 to the 1.

So the general rule is that

the square root of x is x to the half.

And you can generalize this by a very

similar argument that the nth root of x

is x to the (1 over n)

So those are the basic properties

of exponents. Take a moment to

practice them. Try them out in the

next couple of quizzes to make sure

that you see how all of this works

and then we'll move on and think about

logarithms.