>> Welcome to Module 3.
In this module, we're going to be talking
all about exponentials and logarithms,
and we're going to start with exponentials.
What do I mean when I say an exponential?
It's something of that form over here.
We have this a to the x. Great.
Well, basically, the x is
our variable, as always.
Might not be x; it could be
a y, but it's our variable,
and our variable is the exponent in this case.
But we have a as the base.
It's just a number, and that number, as
we can see, it could be a positive number;
it can be a faction; it could be a decimal.
Technically, it could be a negative number.
I didn't show that case here, but it could be.
It's just a is some whole number raised to
an exponent, which is itself a variable.
But how do these look?
Well, if we have a positive number, a
positive number that is greater than 1,
we're going to get what is
known as exponential growth.
Basically, a function that grows
and grows faster and faster,
and you think x again is the exponent.
So, when x is 1, you just
get the same thing back.
When it's 2, you're squaring
it; 3, you're cubing it.
So, our number is going to grow very
rapidly, and if we look at this graph,
I actually compared the different ones.
What we've got is x, or 2x and 3x,
and we see 3x grows much faster.
The bigger the base, the
faster it's going to grow.
But if we get a number that's between 0 and
1, what we actually get is exponential decay.
We're getting less of the number over time,
and that's what this is showing right here.
And in this case, the decay rate,
again, will depend on the fraction.
But what we find is the smaller the fraction,
or the smaller the decimal,
the faster it will decay.
So, that's when you're looking at 0.9 versus
0.5x, you can see 0.5x is going to drop faster.
So, that's what we mean by exponentials;
something, a base number that's a constant,
raised to some exponent, that's a variable.
And now the variable might also
be like 2x, something like that.
We're going to see a bunch
of cases solving those later.
But we also want to talk about logarithms,
and this tends to confuse a lot of people
because you look at a logarithm;
it's a button on your calculator.
You might have heard of it
before, but what does it even mean?
What is it used for?
The main thing to keep in mind is an exponential
and a logarithm are very, very related.
In fact, what do I have here?
I got log to the base a of x
equals y. What does that even mean?
I mean, this is showing us over here;
it's explaining I, but again I'm going
to try to give it in more detail here.
What have we got?
I said it's the same as this, a
to the y equals x. Well, it is.
A logarithm, what it's asking us to do is what
does our base, our a need to be raised to?
What exponent do we need to put
a to in order that we get x?
If I want x, how do I make a into x?
That's why this one is showing that very nicely.
What have we got?
Log to the base 2 of 64.
So, what power does 2 have
to be put to to get 64?
Well, counting it out on their
fingers, or you check on the calculator.
If we go 2 to the power of 6, that equals 64.
That's what the logarithm is all about.
What power do I have to raise a,
so what, and that's what y is.
What power does a have to be
raised to to make it equal to x?
That's what the logarithm is all about.
So, that's, hopefully, at
least now you know what it is.
But I also want to talk about a couple of
special cases, so-called special logarithms,
or mainly kind of confusing ones because
you might see these logarithms; and I said,
normally, you're going to have log to
the base a of x. There's always a base.
But there's a few cases where
you might not see the base.
These so-called special or common logarithms;
in fact, if you look at your calculator,
you probably have a log button
that doesn't have a base.
It's already breaking the
very first rule I told you.
It's got to have a base.
Well, if we ever see just log; oops,
if I could spell log, it would help;
but if we ever just see log of say of
x, it's implied that that's base 10.
That's just a common thing in math that it's
known if the base isn't written, it's base 10,
or someone screwed up and didn't write the base.
But more than likely, if the base is not
there, or if you're using your calculator
and using log, that's log to the base 10.
But it's worth noting that
this base can be anything.
It can be 2, 3, 10, a billion, 1.4762781.
It can be anything we want, and
there's a very common base that's used.
The exponential function, e; our e function,
which is a very important number in math
and science, it's a never-ending number,
just like pie is the one with
infinite digits, e is also.
We can round it off to 2.72 for our purposes.
It technically keeps going on forever, but this
is a very useful number in math and science.
And so, we often have log, now I
don't like those zeros, or the Os.
Log to the base e is a very common one,
and maybe we'll use z in this case.
Log to the base e. Well, this gets
its own special representation
because it's used so often.
If we have log to the base e, that's
the same as saying ln of our variable.
So, if you see ln, it means log to the base e.
If you see just log, it's log to the base 10.
Both of these should be on your calculator.
They're very common ones used.
We're going to use them in the
module, and hopefully you got this.
We'll get you started for Module 3,
and I will see you throughout it.
