## www.mathcentre.ac.uk/.../Exponential%20and%20logarithm%20functions.mp4

• 0:01 - 0:05
In this video, we'll be looking
at exponential functions and
• 0:05 - 0:08
logarithm functions, and I'd
like to start off by thinking
• 0:08 - 0:14
about functions of the form F of
X equals A to the power of X,
• 0:14 - 0:18
where a is representing real
positive numbers. I'm going to
• 0:18 - 0:22
split this up into three cases.
First of all, the case when a
• 0:22 - 0:28
equals 1 hour, then going to
look at the case when A is more
• 0:28 - 0:31
than one, and finally I'll look
at the case where.
• 0:31 - 0:37
Is between zero and one. So
first of all.
• 0:37 - 0:43
When a equals 1, this will give
us the function F of X equals 1
• 0:43 - 0:49
to the power of X and we can see
that once the power of anything
• 0:49 - 0:55
is actually one. So this is the
linear function F of X equals 1.
• 0:55 - 1:00
So that's quite straightforward,
and Secondly, I'd like to look
• 1:00 - 1:04
at the case where a is more than
• 1:04 - 1:09
one. And to demonstrate what
happens in this case, I'd like
• 1:09 - 1:13
to consider a specific example.
In this case, I'll choose A to
• 1:13 - 1:18
be equal to two, which gives us
the function F of X equals 2 to
• 1:18 - 1:20
the power of X.
• 1:21 - 1:24
these kind of functions is to
• 1:24 - 1:27
look at for some different
values of the argument.
• 1:27 - 1:32
So starts off by looking at F of
0, which is actually equal to 2
• 1:32 - 1:34
to the power 0.
• 1:34 - 1:39
And we know that anything to the
power 0 equals 1.
• 1:39 - 1:42
Next, we'll look at F of one.
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Which is 2 to the power of 1.
• 1:45 - 1:47
And two to the power of one is
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2. And we can look at F of
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two. Which is 2 squared,
which is 4. So quite
• 1:59 - 2:02
straightforward, and finally F
• 2:02 - 2:08
of three. Which is 2 to the
power of three which actually
• 2:08 - 2:09
gives us 8.
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Also want to consider some
negative arguments as well, so
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if we look at F of minus one.
• 2:17 - 2:22
This is 2 to the power of
minus one. And remember when
• 2:22 - 2:25
we have a negative power,
that means that we have to
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invert our number so we
actually end up with one
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half.
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If you look at F of minus two
I guess is 2 to the power of
• 2:38 - 2:42
minus two. Once again, this
negative power makes we've got
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one over 2 squared and it's 2
squared is 4's actually gives us
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1/4. And final arguments are
consider is F of minus three.
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Which is 2 to the power of
minus three, which gives us
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one over 2 cubed and two
cubes 8. So we get one 8th.
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I'm going to take these
results now and put them into
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a table and we can use that
table to help us plot a graph
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of the function.
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So our table, the
values of X&F of X.
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We just come from minus 3
- 2 - 1 zero, 1
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two and three and the value
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hardware 1/8.
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1/4
1/2 one, 2,
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four and eight.
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So we're going to plot these now
so we can get a graph of the
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function. So do RF of X
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axis here. And X axis
horizontally. So on the X axis
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we need to go from minus three
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to +3. So minus 1
- 2 - 3 and
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one. Two and three this
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way. And on the vertical scale,
the F of X axis, we need to go
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up to 8. So
12345678. Make
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sure we
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label that.
So now let's plot the
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points.
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Minus 3 1/8.
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Minus 2
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1/4. Minus
1 1/2.
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Zero and one.
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12
214
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And finally, three and eight.
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And so we need to try and draw a
smooth curve through the points.
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And this is the graph
of the function F of
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X equals 2 to the
power of X.
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Now this is actually quite
clearly shows the general shape
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of graphs of the functions where
F of X equals A to the X, and a
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is more than one. However, what
happens when we vary the value
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of A? Well, by looking at a few
sketches of a few different
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graphs, they should become
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clear. So I have my axes again.
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F of X.
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X horizontally.
We've just spotted the graph of
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this function. Which one through
1 F of X axis and this was F of
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X equals 2 to the power of X.
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If we were to look at this.
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Graph.
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This might represent F of X
equals 5 to the power of X.
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If I was look at this graph.
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This might represent the graph
of the function F of X equals 10
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to the power of X.
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So what's actually happening
here? Well, for bigger values of
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a. We can see that the output
increases more quickly as the
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arguments increases.
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Another couple of important
points to notice here are first
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of all that every single graph
that I've sketched here comes
• 7:12 - 7:17
through this .0 one, and in
fact, regardless of our value of
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A. F of 0.
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Will equal 1 for every single
value of a.
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Secondly, we notice that F of
X is always more than 0.
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In other words, are output for
this function is always
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positive. As a very
important feature of these
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kind of functions.
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The last case I would like to
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consider. Is the case where a is
between zero and one case where
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a is between zero and one.
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To demonstrate this case, I
would like it to look at a
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specific example. In this case I
will choose a equals 1/2, so
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this means I'm looking at the
function F of X.
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Equals 1/2 to
the power of X.
• 8:17 - 8:21
Now with the last example, a
good place to start is by
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looking at some different values
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for the arguments. So let's
first of all consider F of 0.
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This will give us 1/2 to the
power of 0.
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And as we said before, anything
to the power of 0 is one.
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Secondly, we look at F of one.
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Which is 1/2 to the power of 1.
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Which is equal to 1/2.
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Half of 2.
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Equals 1/2 squared.
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So on the top that just gives
US1 squared, which is one.
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On the bottom 2 squared, which
is 4. So we end up with one
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quarter. An F of three.
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Is equal to 1/2 cubed.
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So on the top we get one cubed
which just gives US1 and under
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bottom 2 cubed which gives us 8.
So we end up with one 8th.
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And as before, we also need
to consider some negative
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arguments.
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So half of minus one.
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Gives us 1/2 to the power of
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minus one. And remember the
minus sign on the power actually
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inverts are fraction, so we end
up with two over 1 to the power
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of 1, which is just two.
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F of
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minus 2. Gives
us 1/2 to the power of minus
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2. Which is 2 over 1 squared.
You can see on the top we get 2
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squared which is 4 and on the
bottom we just get one. So
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that's actually equal to 4.
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And finally, F of minus three.
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Is one half to the power of
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-3? Which means we get two over
one and we deal with the minus
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sign cubed. So 2 cubed in the
top which is 8 and again just
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the one on the bottom. So that
just gives us 8.
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So once again, we're going to
take these results and put them
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into a table so we can plot a
graph. A graph of the function.
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So X&F of X
again for our table.
• 10:56 - 11:01
And we have values of the
argument ranging from minus
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three, all the way up to
three again.
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And this time the
values where 8421.
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1/2 one quarter.
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And one 8th.
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So let's plot this now
on a graph.
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So vertically we get F of X and
a horizontal axis. We've got X
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and we're going from minus three
to three again.
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So minus 1 - 2 -
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3. 123
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here. And
then we're going up
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to 8 on the
vertical axis, 12345678.
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So let's
plot the
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points. First point
is minus three 8.
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Minus two and four.
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But here. Minus one and two.
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But the. Zero and one.
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One 1/2.
2 one quarter.
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And finally 3 1/8.
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As with our previous example, we
need to try and draw a smooth
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curve through the points.
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So.
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And this represents the function
F of X equals 1/2 to
• 13:04 - 13:06
the power of X.
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And actually this demonstrates
the general shape for functions
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of the form F of X equals A to
the X when A is between zero and
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one. But what happens when we
vary a within those boundaries?
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Well, sketching a few
graphs of this function
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will help us to see.
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Just do some axes again.
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So we've got F of X.
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And X. We've just seen.
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This curve, which was.
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F of X equals 1/2 to
the power of X.
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We might have seen.
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This would have represented.
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F of X equals.
• 14:09 - 14:14
1/5 to the power of X or we may
even have seen.
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Something like this?
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Which would have represented
maybe F of X equals 110th to the
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power of X.
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So we can see that for bigger
values of a this is. We come
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down here. The output decreases
more slowly as the arguments
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increases. And a few important
points to notice here. First of
• 14:47 - 14:51
all, as with the previous
example, F of 0 equals 1
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regardless of the value of A and
in fact, as long as a is
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positive and real, this will
always be the case.
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Second thing to notice, as with
the previous example, is that
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our output F of X is always
positive, so output is always
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more than 0.
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So now we've looked at what
happens for all the different
• 15:19 - 15:22
values of a when A is positive
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and real. No one
interesting thing you
• 15:25 - 15:28
might have noticed is
this. We've got some
• 15:28 - 15:31
symmetry going on here. If
I actually just put some
• 15:31 - 15:33
axes down again here.
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So F of X find X you'll
remember.
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This curve here.
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Passing through the point one
was F of X equals 2 to the power
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of X. And also
this curve here.
• 15:56 - 16:03
Was F of X equals 1/2
to the power of X?
• 16:04 - 16:09
And we can see this link it
'cause one of these graphs is a
• 16:09 - 16:14
reflection of the other in the F
of X axis. And in fact this
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could have been 1/5 to the X and
this could have been 5 to the X
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'cause generally speaking.
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F of X equal to.
• 16:24 - 16:26
AX.
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Is a
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reflection. Of F of
X equals 1 over
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8 to the X
and that is in
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the F of X
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axis. And one over 8 to the
X can also be written as A to
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the minus X.
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So that's an interesting point
• 17:00 - 17:06
to note. Now you might recall
from Chapter 2.3 that the
• 17:06 - 17:08
exponential number E.
• 17:08 - 17:11
Is approximately
• 17:11 - 17:16
2.718. Which means it falls
into the first category, where
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a is more than one. If we're
going to consider the function
• 17:21 - 17:25
F of X equals E to the X, which
is the exponential function.
• 17:28 - 17:30
So if I do some axes again.
• 17:31 - 17:33
F of X vertically.
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An ex horizontally.
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This graph here.
• 17:42 - 17:47
Might represent F of X equals E
to the X and this is what the
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exponential function actually
• 17:48 - 17:53
looks like. I might also like to
consider the function F of X
• 17:53 - 17:56
equals E to the minus X.
• 17:56 - 17:58
As we've just discovered.
• 18:00 - 18:06
That is a reflection in the F of
X axis, so this will be F of X
• 18:06 - 18:11
equals E to the minus X. And
remember this important .1 on
• 18:11 - 18:13
the F of X axis.
• 18:14 - 18:17
So look that exponential
functions and we've looked the
• 18:17 - 18:20
functions of the form A to the
• 18:20 - 18:23
X. At my work to consider
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logarithm functions. And
logarithm functions take the
• 18:27 - 18:33
form F of X equals the log
of X. So particular base. In
• 18:33 - 18:34
this case a.
• 18:36 - 18:39
And as with the previous
example, I'd like to split my
• 18:39 - 18:43
analysis of this into three
parts. First of all, looking at
• 18:43 - 18:44
when a is one.
• 18:45 - 18:49
Second of all, looking at when
there is more than one, and
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finally when a is a number
between zero and one, and as
• 18:53 - 18:57
with the previous example, a is
only going to be positive real
• 18:57 - 19:02
numbers. So first of all, what
happens when a equals 1?
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Well, this means we'll get a
function F of X equals the log
• 19:08 - 19:10
of X to base one.
• 19:11 - 19:14
Remember, this is equivalent.
• 19:15 - 19:17
To say that this number one.
• 19:17 - 19:20
To some power.
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F of X is equal to X, just
like earlier on, this is
• 19:26 - 19:33
equivalent. Generally. 2A
to the power of F of X
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equals X.
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So when we look at this, we can
see that we can only get
• 19:40 - 19:44
solutions when we consider the
arguments X equals 1.
• 19:44 - 19:49
And in fact, if we look at the
arguments X equals 1, there is
• 19:49 - 19:53
an infinite number of answers be
cause one to any power will give
• 19:53 - 19:57
us one. So actually this is not
a valid function because we've
• 19:57 - 20:00
got many outputs for just one
single input. So that's what
• 20:00 - 20:05
happens for a equals 1 would
happen for a is more than one.
• 20:05 - 20:09
Let's look at the case when I is
more than one.
• 20:10 - 20:15
So this means we're looking at
the function F of X equals and
• 20:15 - 20:20
in this case I will choose a
equals 2 again to demonstrate
• 20:20 - 20:26
what's happening. So F of X
equals the log of X to base 2.
• 20:27 - 20:29
And remember, this is
• 20:29 - 20:34
equivalent. The same 2 to the
power of F of X.
• 20:34 - 20:40
Equals X. So as with all
functions, when we get to this
• 20:40 - 20:43
kind of situation, we want to
start looking at some different
• 20:43 - 20:47
values for the argument to help
us plot a graph of the function.
• 20:48 - 20:51
However, before we do that, we
might just want to take a closer
• 20:51 - 20:53
look at what's going on here.
• 20:54 - 21:00
Because 2 to the power of F of
X. In other words, two to some
• 21:00 - 21:05
power can never ever be negative
or 0. This is a positive number.
• 21:05 - 21:09
Whatever we choose, and since
that's more than zero, and
• 21:09 - 21:14
that's equal to X, this means
that X must be more than 0.
• 21:14 - 21:17
And X represents our
arguments, which means I'm
• 21:17 - 21:20
only going to look at
positive arguments for this
• 21:20 - 21:20
reason.
• 21:21 - 21:25
So start off by looking
at F of one.
• 21:26 - 21:30
Now F of one gives us.
• 21:30 - 21:33
The log of one to base 2.
• 21:34 - 21:38
North actually means remember.
• 21:38 - 21:45
Is that two to some power? Half
of one is equal to 1, so two to
• 21:45 - 21:50
what power will give me one?
Well, it must be 2 to the power
• 21:50 - 21:55
zero. Give me one because
anything to the power zero gives
• 21:55 - 22:01
me one. So this half of one must
be 0. So therefore F of one
• 22:01 - 22:07
equals 0. Next I look
at F of two.
• 22:08 - 22:11
And this is means that we've got
the log of two.
• 22:12 - 22:13
To base 2.
• 22:13 - 22:17
And remember that this is
equivalent to saying that two to
• 22:17 - 22:22
some power. In this case F of
two is equal to two SO2 to what
• 22:22 - 22:24
power will give Me 2?
• 22:25 - 22:30
Well, we know that 2 to the
power of one will give Me 2, so
• 22:30 - 22:33
this F of two must be equal to
• 22:33 - 22:40
1. And also going to look
at F of F of four.
• 22:40 - 22:47
Now F of four means that
we've got the log of four
• 22:47 - 22:51
to base two. Remember, this is
• 22:51 - 22:53
equivalent. Just saying that
• 22:53 - 22:58
we've got. Two to some
power. In this case, F
• 22:58 - 23:00
of four is equal to 4.
• 23:01 - 23:04
Now 2 to what power will give me
• 23:04 - 23:10
4? Well, it's actually going to
be 2 squared, so it must be 2 to
• 23:10 - 23:14
the power of two. Will give me
4, so therefore we can see that
• 23:14 - 23:16
F of four will actually give Me
• 23:16 - 23:22
2. I also want to consider
some fractional arguments here
• 23:22 - 23:24
between zero and one.
• 23:25 - 23:28
So if I look at for example.
• 23:28 - 23:31
F of 1/2.
• 23:32 - 23:38
OK, this means we've got the log
of 1/2 to the base 2.
• 23:38 - 23:41
And this is equivalent
to saying.
• 23:43 - 23:50
The two to some power. In this
case F of 1/2 equals 1/2.
• 23:51 - 23:54
Now this time it's a little bit
more tricky to see actually
• 23:54 - 23:55
what's going on here.
• 23:56 - 24:02
But remember, we can write 1/2
as 2 to the power of minus one.
• 24:03 - 24:06
minus power that we talked about
• 24:06 - 24:10
earlier on. So 2 to the power of
sampling equals 2 to the power
• 24:10 - 24:13
of minus one. This something
must be minus one.
• 24:14 - 24:21
So therefore. Half
of 1/2 actually equals minus
• 24:21 - 24:28
one. And the final argument
I want to consider is F of
• 24:28 - 24:34
1/4. And this gives us
the log of 1/4.
• 24:35 - 24:36
So the base 2.
• 24:36 - 24:40
So this is equivalent to saying.
• 24:41 - 24:48
That two to some power. In this
case F of 1/4 is equal to
• 24:48 - 24:53
1/4. And once again, it's not an
easy step just to see exactly
• 24:53 - 24:56
what's going on here. Straight
away. We want to try and rewrite
• 24:56 - 24:58
this right hand side.
• 24:58 - 25:04
Now one over 4 is the same as
one over 2 squared. +2 squared
• 25:04 - 25:09
is just the same as four, and
remember about the minus sign so
• 25:09 - 25:16
we can put that onto the top and
we get 2 to the power of minus
• 25:16 - 25:21
two. So here 2 to the power of
something equals 2 to the power
• 25:21 - 25:26
of minus two. That something
must be minus two, so therefore
• 25:26 - 25:28
F of 1/4 equals minus 2.
• 25:29 - 25:33
So what we're going to do now?
We're going to put these results
• 25:33 - 25:37
into a table so we can plot a
graph of the function.
• 25:38 - 25:44
So we've got
X&F of X.
• 25:44 - 25:51
Now X values we
• 25:51 - 25:54
• 25:55 - 25:59
1, two and four.
• 25:59 - 26:05
And the corresponding outputs
Here were minus 2 - 1
• 26:05 - 26:08
zero one and two.
• 26:08 - 26:12
So let's look at plotting the
graph of this function now, so
• 26:12 - 26:14
as before, we want some axes
• 26:14 - 26:18
here. So F of
• 26:18 - 26:21
X vertically. And X
• 26:21 - 26:28
horizontally. Horizontally we
need to go from
• 26:28 - 26:32
one 3:45, so just
• 26:32 - 26:38
go 1234. Includes everything we
need and vertically we need to
• 26:38 - 26:40
go from minus 2 + 2.
• 26:41 - 26:43
So minus 1 - 2.
• 26:43 - 26:48
I'm one and two, so let's
put the points.
• 26:49 - 26:57
1/4 and minus 2. First of
all, 1/4 minus two 1/2 -
• 26:57 - 27:04
1. So it's 1/2
- 1 one and 0.
• 27:05 - 27:06
21
• 27:07 - 27:11
And finally, four and two.
• 27:12 - 27:16
So now we want to try and
draw a smooth curve through
• 27:16 - 27:19
the point so we can see the
graph of the function.
• 27:20 - 27:28
Excellent, so this
is F of
• 27:28 - 27:36
X equals the
log of X.
• 27:36 - 27:40
These two. And actually, this
represents the general shape for
• 27:40 - 27:45
functions of the form F of X
equals the log of X to base a
• 27:45 - 27:49
when A is more than one. But
what happens as we very well,
• 27:49 - 27:53
let's have a look at a few
sketches of some graphs of some
• 27:53 - 27:57
different functions and that
should help us to see what's
• 27:57 - 28:03
going on. So if I have
F of X vertically.
• 28:03 - 28:07
And X horizontally.
• 28:08 - 28:10
We've just seen.
• 28:11 - 28:15
F of
• 28:15 - 28:21
X equals. Log
of X to base 2.
• 28:22 - 28:24
And if I was to draw this.
• 28:27 - 28:29
This might represent.
• 28:30 - 28:36
F of X equals the log of X
to the base E. Remember E being
• 28:36 - 28:40
the number 2.718, the
exponential number and actually
• 28:40 - 28:44
this is called the natural log
and is sometimes written.
• 28:45 - 28:46
LNX
• 28:47 - 28:50
And finally, it might have.
• 28:53 - 28:57
F of X equals.
• 28:57 - 29:02
Log of AXA Base 5 maybe so we
can see that what's happening
• 29:02 - 29:05
here for bigger values of a.
• 29:06 - 29:09
The output is increasing more
• 29:09 - 29:12
slowly. As the arguments
• 29:12 - 29:17
increases. Now a few important
points to notice here, the first
• 29:17 - 29:22
one. It's a notice this point
here, this one on the X axis.
• 29:23 - 29:26
Regardless of our value of A.
• 29:26 - 29:33
F of one will always be 0.
That's true for all values of a
• 29:33 - 29:37
here. Second thing to notice is
something we touched upon
• 29:37 - 29:40
earlier on is just point about
the arguments always being
• 29:40 - 29:43
positive and we can see this
graphically. Here we see we've
• 29:43 - 29:46
got no points to the left of the
F of X axis.
• 29:47 - 29:51
And so X is always more than
• 29:51 - 29:56
0. The final case
I want to look at.
• 29:57 - 30:03
Is the case worth a is between
zero and one. And to demonstrate
• 30:03 - 30:08
this case I will look at the
specific example where a equals
• 30:08 - 30:14
1/2. And if I equals 1/2, the
function we're going to be
• 30:14 - 30:20
looking at is F of X equals the
log of X to the base 1/2.
• 30:21 - 30:27
So remember we can rewrite this.
This is equivalent to saying
• 30:27 - 30:34
that one half to some power. In
this case F of X is equal
• 30:34 - 30:39
example. We need to think
• 30:39 - 30:42
we're going to consider now.
• 30:42 - 30:48
Because 1/2 to any power will
always give me a positive
• 30:48 - 30:53
number. In other words, 1/2. So
the F of X.
• 30:53 - 30:55
Is always more than 0.
• 30:56 - 31:00
And since this is equal to X,
this means that X is more than
• 31:00 - 31:03
0. And so many going to consider
• 31:03 - 31:08
positive arguments. So first of
all, I can set up F of one.
• 31:09 - 31:15
Half of 1 means that we've
got the log of one to the
• 31:15 - 31:17
base of 1/2.
• 31:18 - 31:20
Remember, this is equivalent.
• 31:21 - 31:28
Just saying.
1/2 to some power. In this case,
• 31:28 - 31:32
F of one is equal to 1.
• 31:34 - 31:38
So how does this work? 1/2 to
some power equals 1. Remember
• 31:38 - 31:42
anything to the power of 0
equals 1, so F of one.
• 31:43 - 31:46
Must be 0.
• 31:46 - 31:52
Secondly,
F of two.
• 31:53 - 31:55
This gives us the log of two.
• 31:56 - 31:58
So the base 1/2.
• 31:58 - 32:00
Remember, this is equivalent to
• 32:00 - 32:07
saying. That we've got 1/2 to
some power. In this case F of
• 32:07 - 32:10
two. Is equal to 2.
• 32:11 - 32:15
So how do we find out what this
powers got to be?
• 32:16 - 32:20
Well, we want to look at
rewriting this number 2, and
• 32:20 - 32:24
this is where the minus negative
powers come in useful.
• 32:24 - 32:30
So we can actually write this as
1/2 to the power of minus one.
• 32:30 - 32:36
If 1/2 to the power of F of two
equals 1/2 to the power of minus
• 32:36 - 32:41
one, then these powers must be
the same, so F of two equals
• 32:41 - 32:47
minus one. After two
equals minus one.
• 32:49 - 32:56
Now look at
F of four.
• 32:56 - 33:01
Half of four gives us the log of
four, so the base 1/2.
• 33:02 - 33:07
Remember, this is equivalent to
saying that one half to some
• 33:07 - 33:11
power. In this case F of four.
• 33:12 - 33:14
Must be equal to 4.
• 33:14 - 33:18
And once again, we need to think
• 33:18 - 33:23
side to get a half so we can see
what the power is.
• 33:23 - 33:26
Now for we know we can write us
• 33:26 - 33:32
2 squared. And then using our
negative powers we can rewrite
• 33:32 - 33:34
this as one half.
• 33:34 - 33:40
So the minus two, so if 1/2 to
the power of F of four equals
• 33:40 - 33:45
1/2 to the power of minus two,
then these powers must be equal.
• 33:45 - 33:47
So F of four equals minus 2.
• 33:48 - 33:54
And also we want to
consider some fractional
• 33:54 - 34:00
arguments. So let's look at
F of 1/2.
• 34:00 - 34:03
Half of 1/2.
• 34:03 - 34:08
Gives us the log of 1/2
to the base 1/2.
• 34:09 - 34:13
So lots of haves and this is
equivalent to saying we've got
• 34:13 - 34:19
1/2 to the power of F of 1/2.
That's going to be equal to 1/2.
• 34:19 - 34:23
This first sight might seem a
little bit complicated, but it's
• 34:23 - 34:28
not at all because 1/2 to some
power to give me 1/2. Well
• 34:28 - 34:33
that's just half to the power of
1, so this F of 1/2 is actually
• 34:33 - 34:34
equal to 1.
• 34:35 - 34:42
And finally, like
to consider F
• 34:42 - 34:49
of 1/4. Which is
equal to the log of 1/4 to
• 34:49 - 34:51
the base of 1/2.
• 34:52 - 34:58
I remember this is equivalent to
writing 1/2 to some power. In
• 34:58 - 35:05
this case F of 1/4 is equal
to 1/4, and again, what we need
• 35:05 - 35:10
to do is just think about
rewriting the right hand side
• 35:10 - 35:16
and actually this is the same as
one over 2 squared.
• 35:16 - 35:19
Which is the same as one half.
• 35:20 - 35:26
to some power equals 1/2
• 35:26 - 35:32
squared. The powers must be
equal, so F of 1/4 must equal 2.
• 35:32 - 35:35
After 1/4 equals 2.
• 35:36 - 35:41
So as usual, put these results
into a table so we can plot a
• 35:41 - 35:43
graph of the function.
• 35:44 - 35:48
It's just the table over here
X&F of X.
• 35:51 - 35:54
Arguments were one
• 35:54 - 36:00
quarter 1/2. 1,
two and four.
• 36:01 - 36:07
And the corresponding outputs.
There were two one 0 -
• 36:07 - 36:14
1 and minus two. So let's
plot these points. So first of
• 36:14 - 36:16
all some axes.
• 36:16 - 36:17
F of X.
• 36:18 - 36:25
X. Vertically, we need
to go from minus 2 + 2, so no
• 36:25 - 36:27
problems minus 1 - 2.
• 36:27 - 36:30
One and two.
• 36:30 - 36:33
And horizontally we need to go
all the way up to four.
• 36:33 - 36:37
So 1 two.
• 36:37 - 36:44
314 Let's
plot the points 1/4 and two.
• 36:45 - 36:49
1/2 and
one.
• 36:51 - 36:53
One and 0.
• 36:54 - 36:57
Two negative one.
• 36:58 - 37:01
Four and negative
• 37:01 - 37:07
2. House before going to try
and draw a smooth curve
• 37:07 - 37:08
through these points.
• 37:16 - 37:23
OK, excellent and this is F of X
equals the log of X to base 1/2.
• 37:23 - 37:27
Actually this demonstrates the
general shape for functions of
• 37:27 - 37:35
the form F of X equals log of
X to the base were a is equal
• 37:35 - 37:37
to the number between zero and
• 37:37 - 37:42
one. But what happens as a
varies within those boundaries?
• 37:42 - 37:47
Well, by looking at the sketch
of a few functions like that, we
• 37:47 - 37:51
should be able to see what's
going on. So just do my axes.
• 37:53 - 37:54
F of X.
• 37:55 - 38:00
And X. And
we've just seen.
• 38:04 - 38:11
F of X equals log
of X. It's a base
• 38:11 - 38:15
1/2. Well, we might have had.
• 38:16 - 38:17
Something that looked like.
• 38:18 - 38:24
This. This might have been
F of X equals.
• 38:24 - 38:28
The log of X so base one
over E. Remember either
• 38:28 - 38:31
exponential number or we
might even have hard.
• 38:35 - 38:39
Something which looked like
this. This might have been F of
• 38:39 - 38:42
X equals. Log of X.
• 38:43 - 38:48
Base 1/5
So what's happening for
• 38:48 - 38:52
different values of a well, we
can see that for the bigger
• 38:52 - 38:57
values of a. The output
decreases more quickly as
• 38:57 - 38:59
the arguments increases.
• 39:01 - 39:05
As a couple of other important
points to notice here as well,
• 39:05 - 39:11
firstly, is this .1 again on the
X axis, and in fact we notice
• 39:11 - 39:16
that F of one equals 0
regardless of our value of A and
• 39:16 - 39:18
that's true for any value.
• 39:19 - 39:23
Secondly, is once again this
• 39:23 - 39:27
arguments we can see
graphically. Once again, the X
• 39:27 - 39:30
has to be more than 0.
• 39:31 - 39:35
So now we know what happens
for all the different values
• 39:35 - 39:38
of a when we considering
logarithm functions.
• 39:39 - 39:43
Once again, you may have
notice some symmetry.
• 39:44 - 39:50
This time the symmetry was
centered on the X axis.
• 39:51 - 39:55
If I actually draw two of
the curves here.
• 40:01 - 40:07
This one might have represented
F of X equals the log of X to
• 40:07 - 40:14
base 2. This will might
represent F of X equals log of X
• 40:14 - 40:19
to the base 1/2, and then forget
very important .1.
• 40:19 - 40:21
And we can see here that
• 40:21 - 40:27
actually. The base two function
is a reflection in the X axis of
• 40:27 - 40:31
the function, which has a base
1/2, and in fact that could have
• 40:31 - 40:37
been five and one, 5th or E and
one over E as the base,
• 40:37 - 40:44
generally speaking. F of X
equals the log of X to
• 40:44 - 40:47
base A. Is
• 40:47 - 40:50
a reflection.
• 40:50 - 40:57
Of.
F of X equals.
• 40:58 - 41:04
Log of X to base one over A That
is in the X axis.
• 41:04 - 41:10
Accent, so now we've looked at
reflections in the X axis in the
• 41:10 - 41:15
F of X axis. We've looked at
exponential functions, and we've
• 41:15 - 41:20
looked at logarithm functions.
The final thing I'd like to look
• 41:20 - 41:26
at in this video is whether
there is a link between these
• 41:26 - 41:31
two functions. Firstly, the
function F of X equals Y to the
• 41:31 - 41:33
X. Remember the exponential
• 41:33 - 41:37
function. And the second
function F of X equals the
• 41:37 - 41:40
natural log of X, which we
mentioned briefly earlier on
• 41:40 - 41:44
now. Remember, this means the
log of X to base A.
• 41:45 - 41:49
Well, good place to start would
be to look at the graphs of the
• 41:49 - 41:54
functions. So I'll do that now.
We've got F of X.
• 41:56 - 42:01
And X. And remember, F of X
equals E to the X.
• 42:05 - 42:10
Run along like this. Is this
important? .1 F of X axis.
• 42:10 - 42:14
Stuff of X equals E to the X.
• 42:15 - 42:17
And if of X equals
a natural log of X.
• 42:18 - 42:20
Came from here.
• 42:21 - 42:25
I'm run along something like
this once again going through
• 42:25 - 42:29
that important .1 on the X axis.
• 42:29 - 42:31
Sex equals and natural log of X.
• 42:31 - 42:37
helpful line to draw in here is
• 42:37 - 42:39
this dotted line.
• 42:41 - 42:42
This dotted line.
• 42:43 - 42:49
Represents the graph of the
linear function F of X equals X.
• 42:49 - 42:55
I want to put that in. We can
see almost immediately that the
• 42:55 - 42:58
exponential function is a
reflection of the natural
• 42:58 - 43:03
logarithm function in the line F
of X equals X. What does that
• 43:03 - 43:07
mean? Well, this is equivalent
to saying that the axes have
• 43:07 - 43:12
been swapped around, so to move
from this function to this
• 43:12 - 43:14
function, all my ex file use.
• 43:14 - 43:20
Have gone to become F of X
values and all my F of X
• 43:20 - 43:24
values have gone to become
X values. In other words,
• 43:24 - 43:27
the inputs and outputs
have been swapped around.
• 43:28 - 43:30
So In summary.
• 43:31 - 43:38
This means that the function F
of X equals E to the X.
• 43:38 - 43:41
Is the inverse?
• 43:42 - 43:48
Of the function F
of X equals a
• 43:48 - 43:52
natural log of X.
Title:
www.mathcentre.ac.uk/.../Exponential%20and%20logarithm%20functions.mp4
Video Language:
English