In this video, we'll be looking
at exponential functions and

logarithm functions, and I'd
like to start off by thinking

about functions of the form F of
X equals A to the power of X,

where a is representing real
positive numbers. I'm going to

split this up into three cases.
First of all, the case when a

equals 1 hour, then going to
look at the case when A is more

than one, and finally I'll look
at the case where.

Is between zero and one. So
first of all.

When a equals 1, this will give
us the function F of X equals 1

to the power of X and we can see
that once the power of anything

is actually one. So this is the
linear function F of X equals 1.

So that's quite straightforward,
and Secondly, I'd like to look

at the case where a is more than

one. And to demonstrate what
happens in this case, I'd like

to consider a specific example.
In this case, I'll choose A to

be equal to two, which gives us
the function F of X equals 2 to

the power of X.

Now a good place to start with
these kind of functions is to

look at for some different
values of the argument.

So starts off by looking at F of
0, which is actually equal to 2

to the power 0.

And we know that anything to the
power 0 equals 1.

Next, we'll look at F of one.

Which is 2 to the power of 1.

And two to the power of one is

2. And we can look at F of

two. Which is 2 squared,
which is 4. So quite

straightforward, and finally F

of three. Which is 2 to the
power of three which actually

gives us 8.

Also want to consider some
negative arguments as well, so

if we look at F of minus one.

This is 2 to the power of
minus one. And remember when

we have a negative power,
that means that we have to

invert our number so we
actually end up with one

half.

If you look at F of minus two
I guess is 2 to the power of

minus two. Once again, this
negative power makes we've got

one over 2 squared and it's 2
squared is 4's actually gives us

1/4. And final arguments are
consider is F of minus three.

Which is 2 to the power of
minus three, which gives us

one over 2 cubed and two
cubes 8. So we get one 8th.

I'm going to take these
results now and put them into

a table and we can use that
table to help us plot a graph

of the function.

So our table, the
values of X&F of X.

We just come from minus 3
- 2 - 1 zero, 1

two and three and the value

hardware 1/8.

1/4
1/2 one, 2,

four and eight.

So we're going to plot these now
so we can get a graph of the

function. So do RF of X

axis here. And X axis
horizontally. So on the X axis

we need to go from minus three

to +3. So minus 1
- 2 - 3 and

one. Two and three this

way. And on the vertical scale,
the F of X axis, we need to go

up to 8. So
12345678. Make

sure we

label that.
So now let's plot the

points.

Minus 3 1/8.

Minus 2

1/4. Minus
1 1/2.

Zero and one.

12
214

And finally, three and eight.

And so we need to try and draw a
smooth curve through the points.

And this is the graph
of the function F of

X equals 2 to the
power of X.

Now this is actually quite
clearly shows the general shape

of graphs of the functions where
F of X equals A to the X, and a

is more than one. However, what
happens when we vary the value

of A? Well, by looking at a few
sketches of a few different

graphs, they should become

clear. So I have my axes again.

F of X.

X horizontally.
We've just spotted the graph of

this function. Which one through
1 F of X axis and this was F of

X equals 2 to the power of X.

If we were to look at this.

Graph.

This might represent F of X
equals 5 to the power of X.

If I was look at this graph.

This might represent the graph
of the function F of X equals 10

to the power of X.

So what's actually happening
here? Well, for bigger values of

a. We can see that the output
increases more quickly as the

arguments increases.

Another couple of important
points to notice here are first

of all that every single graph
that I've sketched here comes

through this .0 one, and in
fact, regardless of our value of

A. F of 0.

Will equal 1 for every single
value of a.

Secondly, we notice that F of
X is always more than 0.

In other words, are output for
this function is always

positive. As a very
important feature of these

kind of functions.

The last case I would like to

consider. Is the case where a is
between zero and one case where

a is between zero and one.

To demonstrate this case, I
would like it to look at a

specific example. In this case I
will choose a equals 1/2, so

this means I'm looking at the
function F of X.

Equals 1/2 to
the power of X.

Now with the last example, a
good place to start is by

looking at some different values

for the arguments. So let's
first of all consider F of 0.

This will give us 1/2 to the
power of 0.

And as we said before, anything
to the power of 0 is one.

Secondly, we look at F of one.

Which is 1/2 to the power of 1.

Which is equal to 1/2.

Half of 2.

Equals 1/2 squared.

So on the top that just gives
US1 squared, which is one.

On the bottom 2 squared, which
is 4. So we end up with one

quarter. An F of three.

Is equal to 1/2 cubed.

So on the top we get one cubed
which just gives US1 and under

bottom 2 cubed which gives us 8.
So we end up with one 8th.

And as before, we also need
to consider some negative

arguments.

So half of minus one.

Gives us 1/2 to the power of

minus one. And remember the
minus sign on the power actually

inverts are fraction, so we end
up with two over 1 to the power

of 1, which is just two.

F of

minus 2. Gives
us 1/2 to the power of minus

2. Which is 2 over 1 squared.
You can see on the top we get 2

squared which is 4 and on the
bottom we just get one. So

that's actually equal to 4.

And finally, F of minus three.

Is one half to the power of

-3? Which means we get two over
one and we deal with the minus

sign cubed. So 2 cubed in the
top which is 8 and again just

the one on the bottom. So that
just gives us 8.

So once again, we're going to
take these results and put them

into a table so we can plot a
graph. A graph of the function.

So X&F of X
again for our table.

And we have values of the
argument ranging from minus

three, all the way up to
three again.

And this time the
values where 8421.

1/2 one quarter.

And one 8th.

So let's plot this now
on a graph.

So vertically we get F of X and
a horizontal axis. We've got X

and we're going from minus three
to three again.

So minus 1 - 2 -

3. 123

here. And
then we're going up

to 8 on the
vertical axis, 12345678.

So let's
plot the

points. First point
is minus three 8.

Which is around about here.

Minus two and four.

But here. Minus one and two.

But the. Zero and one.

One 1/2.
2 one quarter.

And finally 3 1/8.

As with our previous example, we
need to try and draw a smooth

curve through the points.

So.

And this represents the function
F of X equals 1/2 to

the power of X.

And actually this demonstrates
the general shape for functions

of the form F of X equals A to
the X when A is between zero and

one. But what happens when we
vary a within those boundaries?

Well, sketching a few
graphs of this function

will help us to see.

Just do some axes again.

So we've got F of X.

And X. We've just seen.

This curve, which was.

F of X equals 1/2 to
the power of X.

We might have seen.

This would have represented.

F of X equals.

1/5 to the power of X or we may
even have seen.

Something like this?

Which would have represented
maybe F of X equals 110th to the

power of X.

So we can see that for bigger
values of a this is. We come

down here. The output decreases
more slowly as the arguments

increases. And a few important
points to notice here. First of

all, as with the previous
example, F of 0 equals 1

regardless of the value of A and
in fact, as long as a is

positive and real, this will
always be the case.

Second thing to notice, as with
the previous example, is that

our output F of X is always
positive, so output is always

more than 0.

So now we've looked at what
happens for all the different

values of a when A is positive

and real. No one
interesting thing you

might have noticed is
this. We've got some

symmetry going on here. If
I actually just put some

axes down again here.

So F of X find X you'll
remember.

This curve here.

Passing through the point one
was F of X equals 2 to the power

of X. And also
this curve here.

Was F of X equals 1/2
to the power of X?

And we can see this link it
'cause one of these graphs is a

reflection of the other in the F
of X axis. And in fact this

could have been 1/5 to the X and
this could have been 5 to the X

'cause generally speaking.

F of X equal to.

AX.

Is a

reflection. Of F of
X equals 1 over

8 to the X
and that is in

the F of X

axis. And one over 8 to the
X can also be written as A to

the minus X.

So that's an interesting point

to note. Now you might recall
from Chapter 2.3 that the

exponential number E.

Is approximately

2.718. Which means it falls
into the first category, where

a is more than one. If we're
going to consider the function

F of X equals E to the X, which
is the exponential function.

So if I do some axes again.

F of X vertically.

An ex horizontally.

This graph here.

Might represent F of X equals E
to the X and this is what the

exponential function actually

looks like. I might also like to
consider the function F of X

equals E to the minus X.

As we've just discovered.

That is a reflection in the F of
X axis, so this will be F of X

equals E to the minus X. And
remember this important .1 on

the F of X axis.

So look that exponential
functions and we've looked the

functions of the form A to the

X. At my work to consider

logarithm functions. And
logarithm functions take the

form F of X equals the log
of X. So particular base. In

this case a.

And as with the previous
example, I'd like to split my

analysis of this into three
parts. First of all, looking at

when a is one.

Second of all, looking at when
there is more than one, and

finally when a is a number
between zero and one, and as

with the previous example, a is
only going to be positive real

numbers. So first of all, what
happens when a equals 1?

Well, this means we'll get a
function F of X equals the log

of X to base one.

Remember, this is equivalent.

To say that this number one.

To some power.

F of X is equal to X, just
like earlier on, this is

equivalent. Generally. 2A
to the power of F of X

equals X.

So when we look at this, we can
see that we can only get

solutions when we consider the
arguments X equals 1.

And in fact, if we look at the
arguments X equals 1, there is

an infinite number of answers be
cause one to any power will give

us one. So actually this is not
a valid function because we've

got many outputs for just one
single input. So that's what

happens for a equals 1 would
happen for a is more than one.

Let's look at the case when I is
more than one.

So this means we're looking at
the function F of X equals and

in this case I will choose a
equals 2 again to demonstrate

what's happening. So F of X
equals the log of X to base 2.

And remember, this is

equivalent. The same 2 to the
power of F of X.

Equals X. So as with all
functions, when we get to this

kind of situation, we want to
start looking at some different

values for the argument to help
us plot a graph of the function.

However, before we do that, we
might just want to take a closer

look at what's going on here.

Because 2 to the power of F of
X. In other words, two to some

power can never ever be negative
or 0. This is a positive number.

Whatever we choose, and since
that's more than zero, and

that's equal to X, this means
that X must be more than 0.

And X represents our
arguments, which means I'm

only going to look at
positive arguments for this

reason.

So start off by looking
at F of one.

Now F of one gives us.

The log of one to base 2.

North actually means remember.

Is that two to some power? Half
of one is equal to 1, so two to

what power will give me one?
Well, it must be 2 to the power

zero. Give me one because
anything to the power zero gives

me one. So this half of one must
be 0. So therefore F of one

equals 0. Next I look
at F of two.

And this is means that we've got
the log of two.

To base 2.

And remember that this is
equivalent to saying that two to

some power. In this case F of
two is equal to two SO2 to what

power will give Me 2?

Well, we know that 2 to the
power of one will give Me 2, so

this F of two must be equal to

1. And also going to look
at F of F of four.

Now F of four means that
we've got the log of four

to base two. Remember, this is

equivalent. Just saying that

we've got. Two to some
power. In this case, F

of four is equal to 4.

Now 2 to what power will give me

4? Well, it's actually going to
be 2 squared, so it must be 2 to

the power of two. Will give me
4, so therefore we can see that

F of four will actually give Me

2. I also want to consider
some fractional arguments here

between zero and one.

So if I look at for example.

F of 1/2.

OK, this means we've got the log
of 1/2 to the base 2.

And this is equivalent
to saying.

The two to some power. In this
case F of 1/2 equals 1/2.

Now this time it's a little bit
more tricky to see actually

what's going on here.

But remember, we can write 1/2
as 2 to the power of minus one.

Because remember about that
minus power that we talked about

earlier on. So 2 to the power of
sampling equals 2 to the power

of minus one. This something
must be minus one.

So therefore. Half
of 1/2 actually equals minus

one. And the final argument
I want to consider is F of

1/4. And this gives us
the log of 1/4.

So the base 2.

So this is equivalent to saying.

That two to some power. In this
case F of 1/4 is equal to

1/4. And once again, it's not an
easy step just to see exactly

what's going on here. Straight
away. We want to try and rewrite

this right hand side.

Now one over 4 is the same as
one over 2 squared. +2 squared

is just the same as four, and
remember about the minus sign so

we can put that onto the top and
we get 2 to the power of minus

two. So here 2 to the power of
something equals 2 to the power

of minus two. That something
must be minus two, so therefore

F of 1/4 equals minus 2.

So what we're going to do now?
We're going to put these results

into a table so we can plot a
graph of the function.

So we've got
X&F of X.

Now X values we
chose. We had one

quarter. We had one half.

1, two and four.

And the corresponding outputs
Here were minus 2 - 1

zero one and two.

So let's look at plotting the
graph of this function now, so

as before, we want some axes

here. So F of

X vertically. And X

horizontally. Horizontally we
need to go from

one 3:45, so just

go 1234. Includes everything we
need and vertically we need to

go from minus 2 + 2.

So minus 1 - 2.

I'm one and two, so let's
put the points.

1/4 and minus 2. First of
all, 1/4 minus two 1/2 -

1. So it's 1/2
- 1 one and 0.

21

And finally, four and two.

So now we want to try and
draw a smooth curve through

the point so we can see the
graph of the function.

Excellent, so this
is F of

X equals the
log of X.

These two. And actually, this
represents the general shape for

functions of the form F of X
equals the log of X to base a

when A is more than one. But
what happens as we very well,

let's have a look at a few
sketches of some graphs of some

different functions and that
should help us to see what's

going on. So if I have
F of X vertically.

And X horizontally.

We've just seen.

F of

X equals. Log
of X to base 2.

And if I was to draw this.

This might represent.

F of X equals the log of X
to the base E. Remember E being

the number 2.718, the
exponential number and actually

this is called the natural log
and is sometimes written.

LNX

And finally, it might have.

F of X equals.

Log of AXA Base 5 maybe so we
can see that what's happening

here for bigger values of a.

The output is increasing more

slowly. As the arguments

increases. Now a few important
points to notice here, the first

one. It's a notice this point
here, this one on the X axis.

Regardless of our value of A.

F of one will always be 0.
That's true for all values of a

here. Second thing to notice is
something we touched upon

earlier on is just point about
the arguments always being

positive and we can see this
graphically. Here we see we've

got no points to the left of the
F of X axis.

And so X is always more than

0. The final case
I want to look at.

Is the case worth a is between
zero and one. And to demonstrate

this case I will look at the
specific example where a equals

1/2. And if I equals 1/2, the
function we're going to be

looking at is F of X equals the
log of X to the base 1/2.

So remember we can rewrite this.
This is equivalent to saying

that one half to some power. In
this case F of X is equal

to X. Answer the previous
example. We need to think

carefully about which arguments
we're going to consider now.

Because 1/2 to any power will
always give me a positive

number. In other words, 1/2. So
the F of X.

Is always more than 0.

And since this is equal to X,
this means that X is more than

0. And so many going to consider

positive arguments. So first of
all, I can set up F of one.

Half of 1 means that we've
got the log of one to the

base of 1/2.

Remember, this is equivalent.

Just saying.
1/2 to some power. In this case,

F of one is equal to 1.

So how does this work? 1/2 to
some power equals 1. Remember

anything to the power of 0
equals 1, so F of one.

Must be 0.

Secondly,
F of two.

This gives us the log of two.

So the base 1/2.

Remember, this is equivalent to

saying. That we've got 1/2 to
some power. In this case F of

two. Is equal to 2.

So how do we find out what this
powers got to be?

Well, we want to look at
rewriting this number 2, and

this is where the minus negative
powers come in useful.

So we can actually write this as
1/2 to the power of minus one.

If 1/2 to the power of F of two
equals 1/2 to the power of minus

one, then these powers must be
the same, so F of two equals

minus one. After two
equals minus one.

Now look at
F of four.

Half of four gives us the log of
four, so the base 1/2.

Remember, this is equivalent to
saying that one half to some

power. In this case F of four.

Must be equal to 4.

And once again, we need to think
about rewriting this right hand

side to get a half so we can see
what the power is.

Now for we know we can write us

2 squared. And then using our
negative powers we can rewrite

this as one half.

So the minus two, so if 1/2 to
the power of F of four equals

1/2 to the power of minus two,
then these powers must be equal.

So F of four equals minus 2.

And also we want to
consider some fractional

arguments. So let's look at
F of 1/2.

Half of 1/2.

Gives us the log of 1/2
to the base 1/2.

So lots of haves and this is
equivalent to saying we've got

1/2 to the power of F of 1/2.
That's going to be equal to 1/2.

This first sight might seem a
little bit complicated, but it's

not at all because 1/2 to some
power to give me 1/2. Well

that's just half to the power of
1, so this F of 1/2 is actually

equal to 1.

And finally, like
to consider F

of 1/4. Which is
equal to the log of 1/4 to

the base of 1/2.

I remember this is equivalent to
writing 1/2 to some power. In

this case F of 1/4 is equal
to 1/4, and again, what we need

to do is just think about
rewriting the right hand side

and actually this is the same as
one over 2 squared.

Which is the same as one half.

Squad So 1/2
to some power equals 1/2

squared. The powers must be
equal, so F of 1/4 must equal 2.

After 1/4 equals 2.

So as usual, put these results
into a table so we can plot a

graph of the function.

It's just the table over here
X&F of X.

Arguments were one

quarter 1/2. 1,
two and four.

And the corresponding outputs.
There were two one 0 -

1 and minus two. So let's
plot these points. So first of

all some axes.

F of X.

X. Vertically, we need
to go from minus 2 + 2, so no

problems minus 1 - 2.

One and two.

And horizontally we need to go
all the way up to four.

So 1 two.

314 Let's
plot the points 1/4 and two.

1/2 and
one.

One and 0.

Two negative one.

Four and negative

2. House before going to try
and draw a smooth curve

through these points.

OK, excellent and this is F of X
equals the log of X to base 1/2.

Actually this demonstrates the
general shape for functions of

the form F of X equals log of
X to the base were a is equal

to the number between zero and

one. But what happens as a
varies within those boundaries?

Well, by looking at the sketch
of a few functions like that, we

should be able to see what's
going on. So just do my axes.

F of X.

And X. And
we've just seen.

F of X equals log
of X. It's a base

1/2. Well, we might have had.

Something that looked like.

This. This might have been
F of X equals.

The log of X so base one
over E. Remember either

exponential number or we
might even have hard.

Something which looked like
this. This might have been F of

X equals. Log of X.

Base 1/5
So what's happening for

different values of a well, we
can see that for the bigger

values of a. The output
decreases more quickly as

the arguments increases.

As a couple of other important
points to notice here as well,

firstly, is this .1 again on the
X axis, and in fact we notice

that F of one equals 0
regardless of our value of A and

that's true for any value.

Secondly, is once again this
thing about the positive

arguments we can see
graphically. Once again, the X

has to be more than 0.

So now we know what happens
for all the different values

of a when we considering
logarithm functions.

Once again, you may have
notice some symmetry.

This time the symmetry was
centered on the X axis.

If I actually draw two of
the curves here.

This one might have represented
F of X equals the log of X to

base 2. This will might
represent F of X equals log of X

to the base 1/2, and then forget
very important .1.

And we can see here that

actually. The base two function
is a reflection in the X axis of

the function, which has a base
1/2, and in fact that could have

been five and one, 5th or E and
one over E as the base,

generally speaking. F of X
equals the log of X to

base A. Is

a reflection.

Of.
F of X equals.

Log of X to base one over A That
is in the X axis.

Accent, so now we've looked at
reflections in the X axis in the

F of X axis. We've looked at
exponential functions, and we've

looked at logarithm functions.
The final thing I'd like to look

at in this video is whether
there is a link between these

two functions. Firstly, the
function F of X equals Y to the

X. Remember the exponential

function. And the second
function F of X equals the

natural log of X, which we
mentioned briefly earlier on

now. Remember, this means the
log of X to base A.

Well, good place to start would
be to look at the graphs of the

functions. So I'll do that now.
We've got F of X.

And X. And remember, F of X
equals E to the X.

Run along like this. Is this
important? .1 F of X axis.

Stuff of X equals E to the X.

And if of X equals
a natural log of X.

Came from here.

I'm run along something like
this once again going through

that important .1 on the X axis.

Sex equals and natural log of X.

Now instead of link, I think
helpful line to draw in here is

this dotted line.

This dotted line.

Represents the graph of the
linear function F of X equals X.

I want to put that in. We can
see almost immediately that the

exponential function is a
reflection of the natural

logarithm function in the line F
of X equals X. What does that

mean? Well, this is equivalent
to saying that the axes have

been swapped around, so to move
from this function to this

function, all my ex file use.

Have gone to become F of X
values and all my F of X

values have gone to become
X values. In other words,

the inputs and outputs
have been swapped around.

So In summary.

This means that the function F
of X equals E to the X.

Is the inverse?

Of the function F
of X equals a

natural log of X.