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.