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.