PROFESSOR: OK, today's about
inverse functions, which is a
new way to create one function
from another one.
And the reason it's so important
is that we want to--
this, the logarithm, is going to
be the inverse function for
e to the x.
We can't live without e to
the x on one side and the
logarithm on the other side.
So here's the idea of
inverse functions.
Well, here are the
letters we use.
Usually y is f of x.
That's the standard letters.
Then, for the inverse, I'm going
to use f with a minus
one above the line.
And notice though, it will
be x is f inverse of y.
And let me show you
why that is.
Let's just remember what
a function is.
So function like f
is, I take an x--
that's my input--
then the function acts on that
input and produces an output.
At some level, this is what a
function is, a bunch of inputs
and the corresponding outputs.
OK, what's the inverse
function?
You can guess what's coming.
I'll reverse those.
For the inverse function,
y will be the input.
So y is now the input.
What used to be the output
is now the input--
just turning them around.
Then the question is, what
x did it come from?
That x that y came from up here
is the x that it goes to
with the inverse function.
So you see the point?
X and y are just getting
reversed.
Let me do an example, because
that's letters, and we need a
first example.
y is x squared.
So that's my function f of x.
f is this squaring function.
If you give me x equals three,
the output is y equals nine.
Now, what's the inverse
function?
The inverse function, I
want to find x from y.
How do I do that?
I take the square root.
So the inverse function
will be x is--
do you like to write square root
of y or do you like to
write y to the one half power?
Both good--
That's the inverse function of
the other, and, of course,
before we said, if x was
three y was nine.
And, now, if y is nine, then x
will come out to be the square
root of nine, three.
Oh, one small point--
well, not so small.
I was really staying there in
this example with x greater or
equal zero.
I don't know want to allow
x equal minus three.
Well, why not?
Because if I allowed x equal
minus three as one of the
inputs, if I extended the
function x squared
to go for all x's.
So if x equal minus three was
allowed input, then y would be
the same answer, nine.
So I would be getting
nine from both
three and minus three.
And then, in the inverse,
I wouldn't know which
one to go back to.
In the inverse, the input would
be nine, but should the
output be three or
minus three.
So the point is, our functions
have to be--
one-to-one is a kind of nice
expression that gives you the
idea, one x for one y,
one y for one x.
And that means that they're
graphs have to go steadily
upwards or steadily downwards,
but not down and up the way y
equal x squared would if
I went over all x's.
Let me do some more examples
before I come to the reason
for this lecture, which is the
exponential and the logarithm.
And let's just look ahead to
what will come near the end of
the lecture.
We know facts about the
exponential, e to the x, and
those facts, when I look at the
inverse function, give me
some different facts, important
facts still, about
the logarithm.
And here is the most
important one.
That the log of a product of two
numbers is the sum of the
two logarithms, very
important fact.
That simple but important
fact is what
made logarithms famous.
And it was the whole basis
for the slide rule.
Well, do you know what
a slide rule is?
Maybe you haven't
ever seen one?
Probably not.
Everybody had them, and then
suddenly nobody has any.
But the point was, on a slide
rule you had a little stick,
another little stick--
I used to drop the thing--
And you push out log y, and then
the second stick measures
out log capital Y, and then you
read off the answer of a
multiplication.
So you were able to multiply,
but kind of inaccurately.
So--
But that doesn't mean that this
logarithm isn't still important.
It is, just not for
slide rules.
I promised two more examples.
Also this is a chance to
think about functions.
So what about the radius of
a circle r and the area a?
There is a function there.
The input is r, and the
area is pi r squared.
So that's some function of
r, input r, output a.
Now, tell me the inverse
function.
The inverse function, I'm
going to input a, and
I want to get r.
So the input for the inverse
function, it's
going to be like this.
I have to solve that
equation for r.
How do I do that?
I divide by pi.
That gives me r squared.
And then, just as there,
I take the square
root, and I have r.
So that's the inverse
function of--
is that a function of r?
No way.
The input is now a.
This is a function of a.
Divide by pi, take the square
root, and you're back to r.
Let me draw the pictures
that go with that.
Because the graph of a function
and its inverse
function are really
quite neat.
So do you know what the
graph of a equals pi r
squared would look like?
Again, r is only going to be
positive, and, now, area's
only going to be positive.
And the graph of pi r squared
is a parabola.
Say at r equal one, I
reach area equal--
What would be the area
if the radius is one?
Plug it in the formula,
the area is pi.
That's that point.
And those are all the
other points.
OK.
So that's the graph, which
was nothing new.
The new graph is the graph
of the inverse function.
OK.
What's up?
This time the input is now
a, and the output is r.
If the area is 0,
the radius is 0.
If the area is pi--
Oh, look, I'm just going
to take this, and
it's going to go here.
If the area is pi, what's
the radius?
Well, put it in the formula.
If the area is pi, I have
pi over pi, one,
square root's one.
The radius is one, of
course it's one.
One there, so that's a point
on the graph of the inverse
function, of this square
root of a over pi.
And what's the rest
of the graph.
Does it look like that?
No way.
Everything is being flipped,
you could say, Or you could
say, a mirror image just
turned over here.
This thing, which started out
like this, is now a square
root function.
The square root function climbs
and comes around like
that like it's a parabola this
way, because that one was a
parabola that way.
All right, let me go on
to a second example.
But that point that these graphs
just flip over the 45
degree line, it's because y and
x are getting switched.
Let me do the second example.
What about temperature?
We could measure temperature
in Fahrenheit, say f.
Or we can measure it in
centigrade or Celsius, say c.
And what's the function?
If I take f, I want to know c.
The centigrade temperature
is some function of f.
Let me at the same
time draw the
picture so we can remember.
So this is now the
forward function.
I'm creating f first, and then
I'm going to create f inverse.
OK.
Do you remember the point
how they're connected?
Here is f.
And we'll start with the
freezing point of water.
The freezing point of water is
32 Fahrenheit but is zero
centigrade.
That's why that system
got created.
So f equals 32, c equals zero.
That's on the graph.
And then what's the
other key point?
The boiling point of water, so
say, that's one f is 212.
212 is the boiling point
of water in Fahrenheit.
And what's the boiling point
in Celsius, centigrade?
100--
I mean that system was--
I don't know where 32 and 212
came from, but 0 to 100 is
pretty sensible,
0 and then 100.
So that's the other point and
then, actually, the graph is
just straight line.
In fact, let's find
the formula.
What's the equation
for that line?
So I take f and I subtract 32,
so that gets me at the right
start, the right
freezing point.
And now I want to multiply by
the right slope to get up to
the right boiling point.
So when I go over 180,
I want to go up 100.
So it's 100 over 180.
That 180 was the 32 to 212.
So the ratio of 100 to 180
that's, well, five to nine
would be easier to write, so
let me write five to nine.
Is that OK for the graph of
the original function?
This is my function
of f giving me c.
Ready for the inverse
function?
Can you do this with me?
c is now the input.
f is now the output.
What was c equals 0 and
a 100, those were the
key points for c.
f equals 32 and 212 were
they key points for f.
This was on the graph, right?
0 centigrade gives
32 Fahrenheit.
100 centigrade matches 212.
That's on the graph.
And again, it's a
line in between.
Hoo!
My picture isn't so fantastic.
That 212 really should be
higher, and that line should
be steeper.
Let's see that from the
formula for f inverse.
What is going to be the
steepness of the second line?
OK.
Here I've done graphs
with some numbers.
Here I'm going to do algebra.
I mean, the point
of algebra is--
you may have wondered, what
was the point of algebra--
the point is to deal with
all numbers at once.
I could write down some other
numbers like, some in between
number like 122 or something,
probably corresponds to a
centigrade of 50.
But I can't live forever
with numbers.
I need symbols.
That's where letters,
algebra, comes in.
So now, I'm going
to do algebra.
I want to get Fahrenheit
out of centigrade.
I want to solve this
equation for f.
How do you solve for f?
Well, first thing is, get rid of
that 5/9, multiply by 9/5.
So now I have 9/5 of c.
So that 5/9 is now over here.
Now, I have an f minus 32.
I want to bring the 32 over
on to the c side.
It'll come over as a plus.
So I've solved this equation
for f, and that's told me,
what's the inverse function.
And you notice, it is
a straight line.
And what's it's slope
by the way?
Its slope is 9/5, where this
had a slope of 5/9.
That's going to happen,
if you multiply.
And sooner or later, in the
inverse, you have to divide.
So one slope is the reciprocal
of the other slope.
Well, it's especially easy when
we see straight lines.
OK.
Now, are we ready for the real
thing, meaning exponentials?
OK.
So I come back to this board,
which tells me what I'm after.
And raise that a little
and go for it.
So, what am I saying here?
I'm saying that the logarithm
is going to be the inverse
function of e to the x.
And it's called the natural
logarithm, and we use this
letter n for natural.
Although, the truth is, that
it's the only logarithm
I ever think of.
I would freely write L-O-G,
because I would always mean
this natural logarithm.
So I'm defining it as the
inverse and probably a graph
is the way to see what
it looks like.
So I need to graph of e
to the x, and then, a
graph of its inverse.
And then, by the way, since
we're doing calculus, our next
lecture is going to
find derivatives.
We know the derivative
of e to the x.
It's e to the x.
That's the remarkable property
that we started with.
Then we'll find the
derivative of the
log, the inverse function.
And it will come out to be
remarkable too, amazing,
amazing, just what we
needed, in fact.
All right, but let's get an idea
what that log looks like.
I know you've seen logs before,
but now we have this
base e, e to the x that only
comes in calculus.
And let's graph it.
So now my function of x
is e to the x, and I
want to graph it.
This is, of course, y.
Actually, I realize, x
can be negative or
positive, no problem.
But y--
e to the x, always comes
out positive.
The graph is going
to be above--
Here's x.
Let me draw the graph from
0 to one and say,
back to minus one.
Then the graph is going to
be above the axis here.
Let's see, where is it?
When x is 0, what's y?
y is e to the 0th power,
which is one.
e to the 0 is one.
The exponential function starts
at one, right there.
That height is one.
Now when x is one, y is e to the
first power, which is e,
about 2.78.
So maybe up there, somewhere
about here.
So that height is e,
corresponding to one.
And what about when
x is minus one?
Then y is e to the minus one.
e to the minus one is--
that minus says, divide.
It's one over e to the first
power, one over 2.78,
something like 1/3 or so,
something about there.
And now, if I put in the
other points here, the
graph looks like that.
And, actually, the reason I
didn't go beyond x equal one
is that it climbs so fast.
e to the x takes off.
It grows exponentially, if you
can allow me to say that.
Which reminds me, we don't
often say, grows
logarithmically.
Well, let's see what grows
logarithmically means.
It means creeping along.
If e to the x is zipping up
real fast, then the log is
going to go up only slowly.
So I want the inverse
function.
Of course, this graph
continues,
gets very, very small.
It continues up here.
It gets very, very big
but keeps going.
Now, ready for x equals log y.
And remember, I'm going to
draw its picture, and I'm
defining that function as the
inverse function of the one we
have. So I'm not going
to give a new
definition, a new function.
It's defined by being the
inverse function.
That's what it is.
But now we know, from experience
with two graphs, we
know what its graph is
going to look like.
So x is now going to be
this graph, and y is
going to be that one.
So y only is positive.
We can only take the log
of positive numbers.
The log of a negative number,
that's something imaginary,
we're not touching that.
The log can come out positive,
zero, negative.
x could be anything.
Here is x.
Let's put in the
points we know.
They'll be the same three points
as here, but you see
that x axis is now vertical, the
y axis is now horizontal.
I put in these points,
now let me put in--
So what's the thing?
When y is 0, what's x?
Yeah, can you get that one?
What's the log--oh, no.
y doesn't make it to 0.
When y is one, that's
what I meant to say.
When y is one, what's
the log of one?
What is the log of one?
That's a key point here,
and we see it.
We got y equal one
when x is 0.
The logarithm of one is--
so when y is one--
is that right?
Logarithm of one, let's
put in that one.
The logarithm of one is 0.
That's a point on our curve.
That's a point on our curve.
This point flips down
to this point.
Can I just remind myself,
because you saw me hesitating,
that the log of one is 0.
It's nice to have a
couple of numbers.
Then what are the other
ones I want?
I want to know the log
of e, and I want to
know the log of 1/e.
And what are those logarithms?
I could look over here.
They're going to be
one and minus one.
But let's just begin to get
the idea of the log.
The log is the exponent.
That's what you should say
to yourself all the time.
What is the logarithm?
The logarithm is the exponent
in the original.
So here the exponent is one,
so the log is one.
What's the exponent there?
One over e, that's e to
the minus one power.
That's log of e to the
minus one power.
And what is that logarithm?
It's the exponent minus one.
Let me plot those points.
Here is e.
So there is one, here
is e, here is 1/e.
The logarithm of one was 0.
That point's on my graph.
The logarithm of e is one.
This point's on my graph.
The logarithm of 1/e
is minus one.
The curve is coming up like that
but bending down just the
way this curve was bending up.
And if I continue the curve, the
logarithm would get more
and more negative.
It's headed down there, but y
is never allowed to be 0.
Headed up here, what happens?
The log of a million, the
log of a trillion--
I mean, we can deal with the
national debt, just take its
log, because it climbs
so slowly.
Notice it kept climbing.
It doesn't peak off here.
That's a little farther than
I intended to draw it.
That's pretty far out on the
y axis, but not very
high on the x axis.
Logarithms of big numbers
are quite small numbers.
And that's actually
why, as we'll see,
people use log paper.
They draw a log-log graphs.
That's to get big numbers
on to the graph by
dealing with logs.
So that's what I want to say
about the logarithm.
Except, to come back to
these two key facts,
especially the first one.
Can I find space for
that first one?
So y is e to the x, as always.
Capital Y would be e to the
capital X. And now, the
interesting property is what
happens if I multiply.
What happens if I multiply
y times Y?
I have, that's e to the x times
e to the X. That's what
the little y and big y were.
Now we're ready to use the
crucial property of the
exponential curve.
I'm asking you, because
you have to know this.
What is e to the x times
e to the capital X?
Suppose x was two and
capital X was three?
Then I have e times e, e
squared, multiplying e times e
times e, three e's.
What do I have?
I've got e times e, time
e times e time e.
All together five e's are
getting multiplied.
I just add the exponents.
That's the big rule for
the exponential.
If I multiply exponentials,
I add the exponents.
Now, I just want to convert that
to a rule for logarithms.
I'm going to do the
inverse function.
I'm going to take the log of
both sides, and, I hope, we're
going to get the right thing.
The logarithm of this is--
Well, what's the logarithm
of this result?
It's the exponent.
The logarithm of this
number is that.
Just the way the logarithm of e
to that number was the one.
The logarithm of e to that
number was the minus one.
The logarithm of this number is
the exponent x plus capital
X. And finally what
is little x?
Well, don't forget where
it came from.
Little x is the exponent for
y, so little x is log y.
And capital X is
log capital Y.
Bunch of symbols
on that board.
And the last line is the one
that we were shooting for.
The logarithm of y times Y
is the sum of the logs.
Because this guy is
also important--
Maybe I don't even
give a proof.
Because it's intimately related
to this one, why don't
I just see it.
What would be the log
of y squared?
Actually, we already-- here.
If I wanted little y squared,
what should I do?
I can get that answer
from what I've done.
The log of little y squared, I
just take big Y to be the same
as little y, I take big X to be
the same as little x, and
I've got the log of y squared.
Then is x plus x, two x's--
but x is the log of y.
If you square a number, you
only double its log.
You're again seeing why these
numbers can grow very quickly
by squaring and squaring and
squaring, but the logarithms
only grow by multiplying by
two, only going up slowly.
And then the general result
would be for any power, not
just n equals two, not just n
equals a whole number, not
just n equals positive numbers,
but all n, will be--
I'll have n of these--
so I'll have n logarithm of y.
So that's a closely related
property that takes the same y
to different powers.
OK.
Lots of symbols today, but you
had to get that logarithm
function straight before we
can take its derivative.
Can I tell you what
its derivative is?
Would you like to
know in advance?
The derivative--
I don't know if I
should tell you.
The derivative of log y, the
derivative of this log
function, turns out to be 1/y.
Isn't that nice.
A really good answer coming
from this function that we
created as an inverse
function.
And I'll just say here that now
we've created the function.
We've got it.
Then I don't mind if you give it
a different letter, give it
another name.
Well, I hope you keep
its name log.
Most people use that name.
But you could use a
different letter.
I'm perfectly happy for you to
write this as the derivative
of log x is 1/x.
Between that and that, I've
just changed letters.
That was like after the real
thinking of this lecture,
which was the when x was an
input and y was an output, and
I really needed two
different letters.
OK, good, that's inverse
functions.
Thank you.
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