WEBVTT

00:00:07.050 --> 00:00:12.550
PROFESSOR: OK, today's about
inverse functions, which is a

00:00:12.550 --> 00:00:16.020
new way to create one function
from another one.

00:00:16.020 --> 00:00:23.980
And the reason it's so important
is that we want to--

00:00:23.980 --> 00:00:28.810
this, the logarithm, is going to
be the inverse function for

00:00:28.810 --> 00:00:30.270
e to the x.

00:00:30.270 --> 00:00:34.090
We can't live without e to
the x on one side and the

00:00:34.090 --> 00:00:36.380
logarithm on the other side.

00:00:36.380 --> 00:00:39.680
So here's the idea of
inverse functions.

00:00:39.680 --> 00:00:42.730
Well, here are the
letters we use.

00:00:42.730 --> 00:00:45.390
Usually y is f of x.

00:00:45.390 --> 00:00:48.340
That's the standard letters.

00:00:48.340 --> 00:00:54.390
Then, for the inverse, I'm going
to use f with a minus

00:00:54.390 --> 00:00:56.940
one above the line.

00:00:56.940 --> 00:01:01.560
And notice though, it will
be x is f inverse of y.

00:01:01.560 --> 00:01:03.480
And let me show you
why that is.

00:01:03.480 --> 00:01:06.310
Let's just remember what
a function is.

00:01:06.310 --> 00:01:11.180
So function like f
is, I take an x--

00:01:11.180 --> 00:01:12.430
that's my input--

00:01:17.220 --> 00:01:24.585
then the function acts on that
input and produces an output.

00:01:29.980 --> 00:01:37.400
At some level, this is what a
function is, a bunch of inputs

00:01:37.400 --> 00:01:40.300
and the corresponding outputs.

00:01:40.300 --> 00:01:42.890
OK, what's the inverse
function?

00:01:42.890 --> 00:01:44.970
You can guess what's coming.

00:01:44.970 --> 00:01:47.350
I'll reverse those.

00:01:47.350 --> 00:01:52.870
For the inverse function,
y will be the input.

00:01:52.870 --> 00:01:56.500
So y is now the input.

00:01:56.500 --> 00:01:59.710
What used to be the output
is now the input--

00:01:59.710 --> 00:02:01.820
just turning them around.

00:02:01.820 --> 00:02:07.400
Then the question is, what
x did it come from?

00:02:07.400 --> 00:02:13.980
That x that y came from up here
is the x that it goes to

00:02:13.980 --> 00:02:17.020
with the inverse function.

00:02:17.020 --> 00:02:18.370
So you see the point?

00:02:18.370 --> 00:02:21.230
X and y are just getting
reversed.

00:02:21.230 --> 00:02:30.070
Let me do an example, because
that's letters, and we need a

00:02:30.070 --> 00:02:31.580
first example.

00:02:31.580 --> 00:02:35.010
y is x squared.

00:02:35.010 --> 00:02:37.510
So that's my function f of x.

00:02:37.510 --> 00:02:41.140
f is this squaring function.

00:02:41.140 --> 00:02:47.090
If you give me x equals three,
the output is y equals nine.

00:02:47.090 --> 00:02:50.850
Now, what's the inverse
function?

00:02:50.850 --> 00:02:54.770
The inverse function, I
want to find x from y.

00:02:54.770 --> 00:02:55.650
How do I do that?

00:02:55.650 --> 00:02:57.270
I take the square root.

00:02:57.270 --> 00:03:00.590
So the inverse function
will be x is--

00:03:00.590 --> 00:03:03.540
do you like to write square root
of y or do you like to

00:03:03.540 --> 00:03:05.910
write y to the one half power?

00:03:05.910 --> 00:03:07.300
Both good--

00:03:07.300 --> 00:03:12.990
That's the inverse function of
the other, and, of course,

00:03:12.990 --> 00:03:16.930
before we said, if x was
three y was nine.

00:03:16.930 --> 00:03:22.680
And, now, if y is nine, then x
will come out to be the square

00:03:22.680 --> 00:03:26.090
root of nine, three.

00:03:26.090 --> 00:03:28.395
Oh, one small point--
well, not so small.

00:03:32.750 --> 00:03:39.340
I was really staying there in
this example with x greater or

00:03:39.340 --> 00:03:41.410
equal zero.

00:03:41.410 --> 00:03:46.440
I don't know want to allow
x equal minus three.

00:03:46.440 --> 00:03:48.170
Well, why not?

00:03:48.170 --> 00:03:51.130
Because if I allowed x equal
minus three as one of the

00:03:51.130 --> 00:03:55.110
inputs, if I extended the
function x squared

00:03:55.110 --> 00:03:57.840
to go for all x's.

00:03:57.840 --> 00:04:03.440
So if x equal minus three was
allowed input, then y would be

00:04:03.440 --> 00:04:05.140
the same answer, nine.

00:04:05.140 --> 00:04:07.580
So I would be getting
nine from both

00:04:07.580 --> 00:04:09.660
three and minus three.

00:04:09.660 --> 00:04:14.520
And then, in the inverse,
I wouldn't know which

00:04:14.520 --> 00:04:16.279
one to go back to.

00:04:16.279 --> 00:04:18.930
In the inverse, the input would
be nine, but should the

00:04:18.930 --> 00:04:20.959
output be three or
minus three.

00:04:20.959 --> 00:04:26.900
So the point is, our functions
have to be--

00:04:26.900 --> 00:04:31.360
one-to-one is a kind of nice
expression that gives you the

00:04:31.360 --> 00:04:35.710
idea, one x for one y,
one y for one x.

00:04:35.710 --> 00:04:40.730
And that means that they're
graphs have to go steadily

00:04:40.730 --> 00:04:48.680
upwards or steadily downwards,
but not down and up the way y

00:04:48.680 --> 00:04:52.930
equal x squared would if
I went over all x's.

00:04:52.930 --> 00:04:58.520
Let me do some more examples
before I come to the reason

00:04:58.520 --> 00:05:04.750
for this lecture, which is the
exponential and the logarithm.

00:05:04.750 --> 00:05:08.650
And let's just look ahead to
what will come near the end of

00:05:08.650 --> 00:05:09.870
the lecture.

00:05:09.870 --> 00:05:16.060
We know facts about the
exponential, e to the x, and

00:05:16.060 --> 00:05:20.150
those facts, when I look at the
inverse function, give me

00:05:20.150 --> 00:05:24.290
some different facts, important
facts still, about

00:05:24.290 --> 00:05:25.130
the logarithm.

00:05:25.130 --> 00:05:27.200
And here is the most
important one.

00:05:27.200 --> 00:05:33.750
That the log of a product of two
numbers is the sum of the

00:05:33.750 --> 00:05:35.880
two logarithms, very
important fact.

00:05:35.880 --> 00:05:39.710
That simple but important
fact is what

00:05:39.710 --> 00:05:41.970
made logarithms famous.

00:05:41.970 --> 00:05:46.930
And it was the whole basis
for the slide rule.

00:05:46.930 --> 00:05:48.870
Well, do you know what
a slide rule is?

00:05:48.870 --> 00:05:50.340
Maybe you haven't
ever seen one?

00:05:50.340 --> 00:05:52.140
Probably not.

00:05:52.140 --> 00:05:56.600
Everybody had them, and then
suddenly nobody has any.

00:05:56.600 --> 00:05:59.690
But the point was, on a slide
rule you had a little stick,

00:05:59.690 --> 00:06:00.585
another little stick--

00:06:00.585 --> 00:06:02.730
I used to drop the thing--

00:06:02.730 --> 00:06:08.940
And you push out log y, and then
the second stick measures

00:06:08.940 --> 00:06:14.310
out log capital Y, and then you
read off the answer of a

00:06:14.310 --> 00:06:15.180
multiplication.

00:06:15.180 --> 00:06:20.136
So you were able to multiply,
but kind of inaccurately.

00:06:20.136 --> 00:06:21.386
So--

00:06:23.970 --> 00:06:27.110
But that doesn't mean that this

00:06:27.110 --> 00:06:28.800
logarithm isn't still important.

00:06:28.800 --> 00:06:31.500
It is, just not for
slide rules.

00:06:31.500 --> 00:06:33.685
I promised two more examples.

00:06:36.430 --> 00:06:38.820
Also this is a chance to
think about functions.

00:06:38.820 --> 00:06:47.130
So what about the radius of
a circle r and the area a?

00:06:47.130 --> 00:06:49.090
There is a function there.

00:06:49.090 --> 00:06:56.260
The input is r, and the
area is pi r squared.

00:06:56.260 --> 00:07:05.230
So that's some function of
r, input r, output a.

00:07:05.230 --> 00:07:07.850
Now, tell me the inverse
function.

00:07:07.850 --> 00:07:11.470
The inverse function, I'm
going to input a, and

00:07:11.470 --> 00:07:13.120
I want to get r.

00:07:13.120 --> 00:07:16.310
So the input for the inverse
function, it's

00:07:16.310 --> 00:07:17.610
going to be like this.

00:07:17.610 --> 00:07:20.430
I have to solve that
equation for r.

00:07:25.260 --> 00:07:26.140
How do I do that?

00:07:26.140 --> 00:07:27.390
I divide by pi.

00:07:30.440 --> 00:07:33.820
That gives me r squared.

00:07:33.820 --> 00:07:35.970
And then, just as there,
I take the square

00:07:35.970 --> 00:07:37.910
root, and I have r.

00:07:37.910 --> 00:07:41.790
So that's the inverse
function of--

00:07:41.790 --> 00:07:43.940
is that a function of r?

00:07:43.940 --> 00:07:45.200
No way.

00:07:45.200 --> 00:07:46.570
The input is now a.

00:07:46.570 --> 00:07:49.840
This is a function of a.

00:07:49.840 --> 00:07:53.720
Divide by pi, take the square
root, and you're back to r.

00:07:53.720 --> 00:07:56.260
Let me draw the pictures
that go with that.

00:07:56.260 --> 00:07:59.680
Because the graph of a function
and its inverse

00:07:59.680 --> 00:08:02.430
function are really
quite neat.

00:08:02.430 --> 00:08:05.260
So do you know what the
graph of a equals pi r

00:08:05.260 --> 00:08:06.300
squared would look like?

00:08:06.300 --> 00:08:11.190
Again, r is only going to be
positive, and, now, area's

00:08:11.190 --> 00:08:12.710
only going to be positive.

00:08:12.710 --> 00:08:17.470
And the graph of pi r squared
is a parabola.

00:08:17.470 --> 00:08:23.660
Say at r equal one, I
reach area equal--

00:08:23.660 --> 00:08:27.480
What would be the area
if the radius is one?

00:08:27.480 --> 00:08:32.799
Plug it in the formula,
the area is pi.

00:08:32.799 --> 00:08:33.960
That's that point.

00:08:33.960 --> 00:08:35.445
And those are all the
other points.

00:08:35.445 --> 00:08:36.650
OK.

00:08:36.650 --> 00:08:43.309
So that's the graph, which
was nothing new.

00:08:43.309 --> 00:08:46.335
The new graph is the graph
of the inverse function.

00:08:46.335 --> 00:08:46.910
OK.

00:08:46.910 --> 00:08:48.320
What's up?

00:08:48.320 --> 00:08:59.590
This time the input is now
a, and the output is r.

00:08:59.590 --> 00:09:02.900
If the area is 0,
the radius is 0.

00:09:02.900 --> 00:09:05.320
If the area is pi--

00:09:05.320 --> 00:09:07.750
Oh, look, I'm just going
to take this, and

00:09:07.750 --> 00:09:08.830
it's going to go here.

00:09:08.830 --> 00:09:15.300
If the area is pi, what's
the radius?

00:09:15.300 --> 00:09:17.100
Well, put it in the formula.

00:09:17.100 --> 00:09:20.030
If the area is pi, I have
pi over pi, one,

00:09:20.030 --> 00:09:21.090
square root's one.

00:09:21.090 --> 00:09:22.770
The radius is one, of
course it's one.

00:09:26.430 --> 00:09:34.520
One there, so that's a point
on the graph of the inverse

00:09:34.520 --> 00:09:39.050
function, of this square
root of a over pi.

00:09:39.050 --> 00:09:42.850
And what's the rest
of the graph.

00:09:42.850 --> 00:09:43.860
Does it look like that?

00:09:43.860 --> 00:09:45.200
No way.

00:09:45.200 --> 00:09:50.640
Everything is being flipped,
you could say, Or you could

00:09:50.640 --> 00:09:56.860
say, a mirror image just
turned over here.

00:09:56.860 --> 00:10:01.120
This thing, which started out
like this, is now a square

00:10:01.120 --> 00:10:02.220
root function.

00:10:02.220 --> 00:10:05.690
The square root function climbs
and comes around like

00:10:05.690 --> 00:10:11.400
that like it's a parabola this
way, because that one was a

00:10:11.400 --> 00:10:14.390
parabola that way.

00:10:14.390 --> 00:10:17.120
All right, let me go on
to a second example.

00:10:17.120 --> 00:10:22.480
But that point that these graphs
just flip over the 45

00:10:22.480 --> 00:10:26.940
degree line, it's because y and
x are getting switched.

00:10:26.940 --> 00:10:28.346
Let me do the second example.

00:10:31.710 --> 00:10:32.960
What about temperature?

00:10:36.200 --> 00:10:41.170
We could measure temperature
in Fahrenheit, say f.

00:10:41.170 --> 00:10:47.260
Or we can measure it in
centigrade or Celsius, say c.

00:10:47.260 --> 00:10:49.850
And what's the function?

00:10:49.850 --> 00:10:53.950
If I take f, I want to know c.

00:10:53.950 --> 00:10:59.340
The centigrade temperature
is some function of f.

00:10:59.340 --> 00:11:01.030
Let me at the same
time draw the

00:11:01.030 --> 00:11:03.470
picture so we can remember.

00:11:03.470 --> 00:11:06.060
So this is now the
forward function.

00:11:06.060 --> 00:11:09.570
I'm creating f first, and then
I'm going to create f inverse.

00:11:09.570 --> 00:11:10.860
OK.

00:11:10.860 --> 00:11:15.610
Do you remember the point
how they're connected?

00:11:15.610 --> 00:11:18.000
Here is f.

00:11:18.000 --> 00:11:21.050
And we'll start with the
freezing point of water.

00:11:21.050 --> 00:11:26.010
The freezing point of water is
32 Fahrenheit but is zero

00:11:26.010 --> 00:11:26.620
centigrade.

00:11:26.620 --> 00:11:30.410
That's why that system
got created.

00:11:30.410 --> 00:11:35.220
So f equals 32, c equals zero.

00:11:35.220 --> 00:11:36.520
That's on the graph.

00:11:39.110 --> 00:11:41.190
And then what's the
other key point?

00:11:41.190 --> 00:11:48.140
The boiling point of water, so
say, that's one f is 212.

00:11:48.140 --> 00:11:51.750
212 is the boiling point
of water in Fahrenheit.

00:11:51.750 --> 00:11:56.150
And what's the boiling point
in Celsius, centigrade?

00:11:56.150 --> 00:11:57.050
100--

00:11:57.050 --> 00:11:58.860
I mean that system was--

00:11:58.860 --> 00:12:03.440
I don't know where 32 and 212
came from, but 0 to 100 is

00:12:03.440 --> 00:12:08.470
pretty sensible,
0 and then 100.

00:12:08.470 --> 00:12:15.880
So that's the other point and
then, actually, the graph is

00:12:15.880 --> 00:12:17.990
just straight line.

00:12:22.150 --> 00:12:26.010
In fact, let's find
the formula.

00:12:26.010 --> 00:12:28.950
What's the equation
for that line?

00:12:28.950 --> 00:12:36.060
So I take f and I subtract 32,
so that gets me at the right

00:12:36.060 --> 00:12:38.330
start, the right
freezing point.

00:12:38.330 --> 00:12:42.810
And now I want to multiply by
the right slope to get up to

00:12:42.810 --> 00:12:44.850
the right boiling point.

00:12:44.850 --> 00:12:48.840
So when I go over 180,
I want to go up 100.

00:12:48.840 --> 00:12:52.400
So it's 100 over 180.

00:12:52.400 --> 00:12:57.200
That 180 was the 32 to 212.

00:12:57.200 --> 00:13:02.040
So the ratio of 100 to 180
that's, well, five to nine

00:13:02.040 --> 00:13:08.090
would be easier to write, so
let me write five to nine.

00:13:08.090 --> 00:13:12.680
Is that OK for the graph of
the original function?

00:13:12.680 --> 00:13:16.960
This is my function
of f giving me c.

00:13:16.960 --> 00:13:24.620
Ready for the inverse
function?

00:13:24.620 --> 00:13:26.700
Can you do this with me?

00:13:26.700 --> 00:13:29.950
c is now the input.

00:13:29.950 --> 00:13:32.600
f is now the output.

00:13:32.600 --> 00:13:36.090
What was c equals 0 and
a 100, those were the

00:13:36.090 --> 00:13:39.920
key points for c.

00:13:39.920 --> 00:13:48.310
f equals 32 and 212 were
they key points for f.

00:13:48.310 --> 00:13:50.780
This was on the graph, right?

00:13:50.780 --> 00:13:54.500
0 centigrade gives
32 Fahrenheit.

00:13:54.500 --> 00:13:56.850
100 centigrade matches 212.

00:13:56.850 --> 00:13:59.630
That's on the graph.

00:13:59.630 --> 00:14:03.967
And again, it's a
line in between.

00:14:03.967 --> 00:14:05.217
Hoo!

00:14:06.930 --> 00:14:09.310
My picture isn't so fantastic.

00:14:09.310 --> 00:14:12.110
That 212 really should be
higher, and that line should

00:14:12.110 --> 00:14:14.810
be steeper.

00:14:14.810 --> 00:14:19.500
Let's see that from the
formula for f inverse.

00:14:19.500 --> 00:14:23.325
What is going to be the
steepness of the second line?

00:14:23.325 --> 00:14:24.575
OK.

00:14:29.300 --> 00:14:33.820
Here I've done graphs
with some numbers.

00:14:33.820 --> 00:14:35.220
Here I'm going to do algebra.

00:14:37.830 --> 00:14:41.620
I mean, the point
of algebra is--

00:14:41.620 --> 00:14:44.510
you may have wondered, what
was the point of algebra--

00:14:44.510 --> 00:14:48.285
the point is to deal with
all numbers at once.

00:14:50.860 --> 00:14:55.020
I could write down some other
numbers like, some in between

00:14:55.020 --> 00:15:00.160
number like 122 or something,
probably corresponds to a

00:15:00.160 --> 00:15:03.420
centigrade of 50.

00:15:03.420 --> 00:15:07.880
But I can't live forever
with numbers.

00:15:07.880 --> 00:15:11.330
I need symbols.

00:15:11.330 --> 00:15:13.560
That's where letters,
algebra, comes in.

00:15:13.560 --> 00:15:15.770
So now, I'm going
to do algebra.

00:15:15.770 --> 00:15:20.960
I want to get Fahrenheit
out of centigrade.

00:15:20.960 --> 00:15:23.940
I want to solve this
equation for f.

00:15:23.940 --> 00:15:25.160
How do you solve for f?

00:15:25.160 --> 00:15:30.920
Well, first thing is, get rid of
that 5/9, multiply by 9/5.

00:15:30.920 --> 00:15:35.280
So now I have 9/5 of c.

00:15:35.280 --> 00:15:38.480
So that 5/9 is now over here.

00:15:38.480 --> 00:15:40.330
Now, I have an f minus 32.

00:15:40.330 --> 00:15:43.500
I want to bring the 32 over
on to the c side.

00:15:43.500 --> 00:15:44.885
It'll come over as a plus.

00:15:48.630 --> 00:15:52.980
So I've solved this equation
for f, and that's told me,

00:15:52.980 --> 00:15:54.340
what's the inverse function.

00:15:54.340 --> 00:15:56.620
And you notice, it is
a straight line.

00:15:56.620 --> 00:16:00.010
And what's it's slope
by the way?

00:16:00.010 --> 00:16:05.630
Its slope is 9/5, where this
had a slope of 5/9.

00:16:05.630 --> 00:16:10.120
That's going to happen,
if you multiply.

00:16:10.120 --> 00:16:14.570
And sooner or later, in the
inverse, you have to divide.

00:16:14.570 --> 00:16:18.930
So one slope is the reciprocal
of the other slope.

00:16:18.930 --> 00:16:21.670
Well, it's especially easy when
we see straight lines.

00:16:21.670 --> 00:16:23.860
OK.

00:16:23.860 --> 00:16:29.670
Now, are we ready for the real
thing, meaning exponentials?

00:16:29.670 --> 00:16:31.170
OK.

00:16:31.170 --> 00:16:35.830
So I come back to this board,
which tells me what I'm after.

00:16:35.830 --> 00:16:40.280
And raise that a little
and go for it.

00:16:43.880 --> 00:16:46.360
So, what am I saying here?

00:16:46.360 --> 00:16:51.950
I'm saying that the logarithm
is going to be the inverse

00:16:51.950 --> 00:16:56.310
function of e to the x.

00:16:56.310 --> 00:16:59.510
And it's called the natural
logarithm, and we use this

00:16:59.510 --> 00:17:03.055
letter n for natural.

00:17:06.170 --> 00:17:10.170
Although, the truth is, that
it's the only logarithm

00:17:10.170 --> 00:17:11.900
I ever think of.

00:17:11.900 --> 00:17:18.349
I would freely write L-O-G,
because I would always mean

00:17:18.349 --> 00:17:20.640
this natural logarithm.

00:17:25.450 --> 00:17:29.600
So I'm defining it as the
inverse and probably a graph

00:17:29.600 --> 00:17:35.560
is the way to see what
it looks like.

00:17:35.560 --> 00:17:39.570
So I need to graph of e
to the x, and then, a

00:17:39.570 --> 00:17:40.660
graph of its inverse.

00:17:40.660 --> 00:17:45.870
And then, by the way, since
we're doing calculus, our next

00:17:45.870 --> 00:17:50.340
lecture is going to
find derivatives.

00:17:50.340 --> 00:17:52.920
We know the derivative
of e to the x.

00:17:52.920 --> 00:17:54.340
It's e to the x.

00:17:54.340 --> 00:18:03.440
That's the remarkable property
that we started with.

00:18:03.440 --> 00:18:07.280
Then we'll find the
derivative of the

00:18:07.280 --> 00:18:10.010
log, the inverse function.

00:18:10.010 --> 00:18:14.060
And it will come out to be
remarkable too, amazing,

00:18:14.060 --> 00:18:17.680
amazing, just what we
needed, in fact.

00:18:17.680 --> 00:18:20.640
All right, but let's get an idea
what that log looks like.

00:18:20.640 --> 00:18:25.070
I know you've seen logs before,
but now we have this

00:18:25.070 --> 00:18:30.890
base e, e to the x that only
comes in calculus.

00:18:30.890 --> 00:18:35.980
And let's graph it.

00:18:35.980 --> 00:18:41.810
So now my function of x
is e to the x, and I

00:18:41.810 --> 00:18:43.060
want to graph it.

00:18:46.070 --> 00:18:49.380
This is, of course, y.

00:18:49.380 --> 00:18:57.220
Actually, I realize, x
can be negative or

00:18:57.220 --> 00:19:00.760
positive, no problem.

00:19:00.760 --> 00:19:02.530
But y--

00:19:02.530 --> 00:19:05.810
e to the x, always comes
out positive.

00:19:05.810 --> 00:19:09.090
The graph is going
to be above--

00:19:09.090 --> 00:19:11.120
Here's x.

00:19:11.120 --> 00:19:14.560
Let me draw the graph from
0 to one and say,

00:19:14.560 --> 00:19:15.810
back to minus one.

00:19:18.980 --> 00:19:25.040
Then the graph is going to
be above the axis here.

00:19:25.040 --> 00:19:26.760
Let's see, where is it?

00:19:30.140 --> 00:19:33.240
When x is 0, what's y?

00:19:33.240 --> 00:19:39.970
y is e to the 0th power,
which is one.

00:19:39.970 --> 00:19:41.890
e to the 0 is one.

00:19:41.890 --> 00:19:45.880
The exponential function starts
at one, right there.

00:19:45.880 --> 00:19:48.960
That height is one.

00:19:48.960 --> 00:19:57.210
Now when x is one, y is e to the
first power, which is e,

00:19:57.210 --> 00:19:59.880
about 2.78.

00:19:59.880 --> 00:20:05.000
So maybe up there, somewhere
about here.

00:20:05.000 --> 00:20:10.600
So that height is e,
corresponding to one.

00:20:10.600 --> 00:20:14.520
And what about when
x is minus one?

00:20:14.520 --> 00:20:19.160
Then y is e to the minus one.

00:20:19.160 --> 00:20:21.650
e to the minus one is--

00:20:21.650 --> 00:20:24.340
that minus says, divide.

00:20:24.340 --> 00:20:29.370
It's one over e to the first
power, one over 2.78,

00:20:29.370 --> 00:20:33.330
something like 1/3 or so,
something about there.

00:20:33.330 --> 00:20:41.290
And now, if I put in the
other points here, the

00:20:41.290 --> 00:20:44.225
graph looks like that.

00:20:44.225 --> 00:20:48.220
And, actually, the reason I
didn't go beyond x equal one

00:20:48.220 --> 00:20:58.450
is that it climbs so fast.
e to the x takes off.

00:20:58.450 --> 00:21:04.990
It grows exponentially, if you
can allow me to say that.

00:21:04.990 --> 00:21:08.710
Which reminds me, we don't
often say, grows

00:21:08.710 --> 00:21:11.340
logarithmically.

00:21:11.340 --> 00:21:14.320
Well, let's see what grows
logarithmically means.

00:21:14.320 --> 00:21:16.770
It means creeping along.

00:21:16.770 --> 00:21:22.220
If e to the x is zipping up
real fast, then the log is

00:21:22.220 --> 00:21:24.550
going to go up only slowly.

00:21:28.230 --> 00:21:29.720
So I want the inverse
function.

00:21:29.720 --> 00:21:33.060
Of course, this graph
continues,

00:21:33.060 --> 00:21:35.250
gets very, very small.

00:21:35.250 --> 00:21:36.830
It continues up here.

00:21:36.830 --> 00:21:42.380
It gets very, very big
but keeps going.

00:21:42.380 --> 00:21:48.700
Now, ready for x equals log y.

00:21:48.700 --> 00:21:55.130
And remember, I'm going to
draw its picture, and I'm

00:21:55.130 --> 00:22:01.430
defining that function as the
inverse function of the one we

00:22:01.430 --> 00:22:04.940
have. So I'm not going
to give a new

00:22:04.940 --> 00:22:07.750
definition, a new function.

00:22:07.750 --> 00:22:17.470
It's defined by being the
inverse function.

00:22:17.470 --> 00:22:19.460
That's what it is.

00:22:19.460 --> 00:22:23.370
But now we know, from experience
with two graphs, we

00:22:23.370 --> 00:22:26.170
know what its graph is
going to look like.

00:22:26.170 --> 00:22:29.130
So x is now going to be
this graph, and y is

00:22:29.130 --> 00:22:29.730
going to be that one.

00:22:29.730 --> 00:22:32.150
So y only is positive.

00:22:32.150 --> 00:22:35.910
We can only take the log
of positive numbers.

00:22:38.680 --> 00:22:41.270
The log of a negative number,
that's something imaginary,

00:22:41.270 --> 00:22:42.520
we're not touching that.

00:22:45.400 --> 00:22:55.990
The log can come out positive,
zero, negative.

00:22:55.990 --> 00:22:57.150
x could be anything.

00:22:57.150 --> 00:23:00.290
Here is x.

00:23:00.290 --> 00:23:02.630
Let's put in the
points we know.

00:23:02.630 --> 00:23:08.020
They'll be the same three points
as here, but you see

00:23:08.020 --> 00:23:15.210
that x axis is now vertical, the
y axis is now horizontal.

00:23:15.210 --> 00:23:18.330
I put in these points,
now let me put in--

00:23:18.330 --> 00:23:21.050
So what's the thing?

00:23:21.050 --> 00:23:26.370
When y is 0, what's x?

00:23:26.370 --> 00:23:28.490
Yeah, can you get that one?

00:23:28.490 --> 00:23:29.740
What's the log--oh, no.

00:23:32.620 --> 00:23:34.690
y doesn't make it to 0.

00:23:34.690 --> 00:23:38.020
When y is one, that's
what I meant to say.

00:23:38.020 --> 00:23:43.250
When y is one, what's
the log of one?

00:23:43.250 --> 00:23:44.810
What is the log of one?

00:23:44.810 --> 00:23:47.730
That's a key point here,
and we see it.

00:23:47.730 --> 00:23:50.580
We got y equal one
when x is 0.

00:23:50.580 --> 00:23:56.930
The logarithm of one is--

00:23:56.930 --> 00:23:59.980
so when y is one--

00:23:59.980 --> 00:24:01.540
is that right?

00:24:01.540 --> 00:24:05.170
Logarithm of one, let's
put in that one.

00:24:05.170 --> 00:24:07.410
The logarithm of one is 0.

00:24:07.410 --> 00:24:10.150
That's a point on our curve.

00:24:10.150 --> 00:24:11.770
That's a point on our curve.

00:24:11.770 --> 00:24:16.690
This point flips down
to this point.

00:24:16.690 --> 00:24:20.620
Can I just remind myself,
because you saw me hesitating,

00:24:20.620 --> 00:24:23.590
that the log of one is 0.

00:24:23.590 --> 00:24:26.910
It's nice to have a
couple of numbers.

00:24:26.910 --> 00:24:28.460
Then what are the other
ones I want?

00:24:28.460 --> 00:24:31.260
I want to know the log
of e, and I want to

00:24:31.260 --> 00:24:34.980
know the log of 1/e.

00:24:34.980 --> 00:24:38.570
And what are those logarithms?

00:24:38.570 --> 00:24:41.160
I could look over here.

00:24:41.160 --> 00:24:44.860
They're going to be
one and minus one.

00:24:44.860 --> 00:24:49.000
But let's just begin to get
the idea of the log.

00:24:49.000 --> 00:24:54.390
The log is the exponent.

00:24:54.390 --> 00:24:56.860
That's what you should say
to yourself all the time.

00:24:56.860 --> 00:24:58.290
What is the logarithm?

00:24:58.290 --> 00:25:03.360
The logarithm is the exponent
in the original.

00:25:03.360 --> 00:25:08.510
So here the exponent is one,
so the log is one.

00:25:08.510 --> 00:25:10.930
What's the exponent there?

00:25:10.930 --> 00:25:14.480
One over e, that's e to
the minus one power.

00:25:14.480 --> 00:25:18.260
That's log of e to the
minus one power.

00:25:18.260 --> 00:25:19.890
And what is that logarithm?

00:25:19.890 --> 00:25:23.260
It's the exponent minus one.

00:25:23.260 --> 00:25:25.030
Let me plot those points.

00:25:25.030 --> 00:25:27.030
Here is e.

00:25:27.030 --> 00:25:30.350
So there is one, here
is e, here is 1/e.

00:25:34.760 --> 00:25:36.750
The logarithm of one was 0.

00:25:36.750 --> 00:25:38.430
That point's on my graph.

00:25:38.430 --> 00:25:40.800
The logarithm of e is one.

00:25:40.800 --> 00:25:42.750
This point's on my graph.

00:25:42.750 --> 00:25:45.760
The logarithm of 1/e
is minus one.

00:25:45.760 --> 00:25:55.040
The curve is coming up like that
but bending down just the

00:25:55.040 --> 00:25:59.420
way this curve was bending up.

00:25:59.420 --> 00:26:03.970
And if I continue the curve, the
logarithm would get more

00:26:03.970 --> 00:26:04.720
and more negative.

00:26:04.720 --> 00:26:11.540
It's headed down there, but y
is never allowed to be 0.

00:26:11.540 --> 00:26:14.320
Headed up here, what happens?

00:26:14.320 --> 00:26:18.450
The log of a million, the
log of a trillion--

00:26:18.450 --> 00:26:22.960
I mean, we can deal with the
national debt, just take its

00:26:22.960 --> 00:26:31.260
log, because it climbs
so slowly.

00:26:31.260 --> 00:26:32.770
Notice it kept climbing.

00:26:32.770 --> 00:26:37.700
It doesn't peak off here.

00:26:37.700 --> 00:26:40.630
That's a little farther than
I intended to draw it.

00:26:40.630 --> 00:26:45.090
That's pretty far out on the
y axis, but not very

00:26:45.090 --> 00:26:46.420
high on the x axis.

00:26:46.420 --> 00:26:50.810
Logarithms of big numbers
are quite small numbers.

00:26:50.810 --> 00:26:56.680
And that's actually
why, as we'll see,

00:26:56.680 --> 00:26:58.490
people use log paper.

00:26:58.490 --> 00:27:00.640
They draw a log-log graphs.

00:27:00.640 --> 00:27:04.570
That's to get big numbers
on to the graph by

00:27:04.570 --> 00:27:06.480
dealing with logs.

00:27:06.480 --> 00:27:12.210
So that's what I want to say
about the logarithm.

00:27:12.210 --> 00:27:20.070
Except, to come back to
these two key facts,

00:27:20.070 --> 00:27:23.010
especially the first one.

00:27:23.010 --> 00:27:26.780
Can I find space for
that first one?

00:27:32.990 --> 00:27:36.130
So y is e to the x, as always.

00:27:36.130 --> 00:27:46.770
Capital Y would be e to the
capital X. And now, the

00:27:46.770 --> 00:27:50.450
interesting property is what
happens if I multiply.

00:27:50.450 --> 00:27:53.010
What happens if I multiply
y times Y?

00:27:55.870 --> 00:28:01.780
I have, that's e to the x times
e to the X. That's what

00:28:01.780 --> 00:28:04.210
the little y and big y were.

00:28:04.210 --> 00:28:09.930
Now we're ready to use the
crucial property of the

00:28:09.930 --> 00:28:11.180
exponential curve.

00:28:13.660 --> 00:28:16.670
I'm asking you, because
you have to know this.

00:28:16.670 --> 00:28:19.905
What is e to the x times
e to the capital X?

00:28:25.520 --> 00:28:29.440
Suppose x was two and
capital X was three?

00:28:29.440 --> 00:28:35.590
Then I have e times e, e
squared, multiplying e times e

00:28:35.590 --> 00:28:39.570
times e, three e's.

00:28:39.570 --> 00:28:40.370
What do I have?

00:28:40.370 --> 00:28:45.520
I've got e times e, time
e times e time e.

00:28:45.520 --> 00:28:47.715
All together five e's are
getting multiplied.

00:28:50.420 --> 00:28:55.440
I just add the exponents.

00:28:55.440 --> 00:28:58.870
That's the big rule for
the exponential.

00:28:58.870 --> 00:29:04.360
If I multiply exponentials,
I add the exponents.

00:29:04.360 --> 00:29:09.040
Now, I just want to convert that
to a rule for logarithms.

00:29:09.040 --> 00:29:10.950
I'm going to do the
inverse function.

00:29:10.950 --> 00:29:14.990
I'm going to take the log of
both sides, and, I hope, we're

00:29:14.990 --> 00:29:16.620
going to get the right thing.

00:29:16.620 --> 00:29:22.790
The logarithm of this is--

00:29:22.790 --> 00:29:27.910
Well, what's the logarithm
of this result?

00:29:27.910 --> 00:29:30.610
It's the exponent.

00:29:30.610 --> 00:29:34.820
The logarithm of this
number is that.

00:29:34.820 --> 00:29:39.120
Just the way the logarithm of e
to that number was the one.

00:29:39.120 --> 00:29:42.260
The logarithm of e to that
number was the minus one.

00:29:42.260 --> 00:29:47.470
The logarithm of this number is
the exponent x plus capital

00:29:47.470 --> 00:29:53.380
X. And finally what
is little x?

00:29:53.380 --> 00:29:55.740
Well, don't forget where
it came from.

00:29:55.740 --> 00:30:02.480
Little x is the exponent for
y, so little x is log y.

00:30:02.480 --> 00:30:09.810
And capital X is
log capital Y.

00:30:09.810 --> 00:30:12.900
Bunch of symbols
on that board.

00:30:12.900 --> 00:30:16.670
And the last line is the one
that we were shooting for.

00:30:16.670 --> 00:30:21.730
The logarithm of y times Y
is the sum of the logs.

00:30:25.360 --> 00:30:27.585
Because this guy is
also important--

00:30:31.940 --> 00:30:35.570
Maybe I don't even
give a proof.

00:30:35.570 --> 00:30:39.150
Because it's intimately related
to this one, why don't

00:30:39.150 --> 00:30:40.720
I just see it.

00:30:40.720 --> 00:30:42.545
What would be the log
of y squared?

00:30:51.370 --> 00:30:53.650
Actually, we already-- here.

00:30:53.650 --> 00:30:57.750
If I wanted little y squared,
what should I do?

00:30:57.750 --> 00:31:00.110
I can get that answer
from what I've done.

00:31:00.110 --> 00:31:04.660
The log of little y squared, I
just take big Y to be the same

00:31:04.660 --> 00:31:08.180
as little y, I take big X to be
the same as little x, and

00:31:08.180 --> 00:31:10.190
I've got the log of y squared.

00:31:10.190 --> 00:31:14.030
Then is x plus x, two x's--

00:31:16.910 --> 00:31:18.810
but x is the log of y.

00:31:23.660 --> 00:31:29.530
If you square a number, you
only double its log.

00:31:29.530 --> 00:31:37.860
You're again seeing why these
numbers can grow very quickly

00:31:37.860 --> 00:31:41.710
by squaring and squaring and
squaring, but the logarithms

00:31:41.710 --> 00:31:46.940
only grow by multiplying by
two, only going up slowly.

00:31:46.940 --> 00:31:54.150
And then the general result
would be for any power, not

00:31:54.150 --> 00:31:57.840
just n equals two, not just n
equals a whole number, not

00:31:57.840 --> 00:32:05.450
just n equals positive numbers,
but all n, will be--

00:32:05.450 --> 00:32:07.670
I'll have n of these--

00:32:07.670 --> 00:32:11.470
so I'll have n logarithm of y.

00:32:11.470 --> 00:32:19.060
So that's a closely related
property that takes the same y

00:32:19.060 --> 00:32:19.740
to different powers.

00:32:19.740 --> 00:32:20.990
OK.

00:32:23.720 --> 00:32:29.170
Lots of symbols today, but you
had to get that logarithm

00:32:29.170 --> 00:32:32.420
function straight before we
can take its derivative.

00:32:32.420 --> 00:32:34.800
Can I tell you what
its derivative is?

00:32:34.800 --> 00:32:37.410
Would you like to
know in advance?

00:32:37.410 --> 00:32:38.660
The derivative--

00:32:41.460 --> 00:32:43.080
I don't know if I
should tell you.

00:32:43.080 --> 00:32:49.160
The derivative of log y, the
derivative of this log

00:32:49.160 --> 00:32:52.600
function, turns out to be 1/y.

00:32:56.730 --> 00:32:58.360
Isn't that nice.

00:32:58.360 --> 00:33:01.950
A really good answer coming
from this function that we

00:33:01.950 --> 00:33:04.040
created as an inverse
function.

00:33:04.040 --> 00:33:06.750
And I'll just say here that now

00:33:06.750 --> 00:33:09.140
we've created the function.

00:33:09.140 --> 00:33:11.220
We've got it.

00:33:11.220 --> 00:33:15.680
Then I don't mind if you give it
a different letter, give it

00:33:15.680 --> 00:33:16.850
another name.

00:33:16.850 --> 00:33:19.590
Well, I hope you keep
its name log.

00:33:19.590 --> 00:33:21.190
Most people use that name.

00:33:21.190 --> 00:33:22.810
But you could use a
different letter.

00:33:22.810 --> 00:33:26.910
I'm perfectly happy for you to
write this as the derivative

00:33:26.910 --> 00:33:31.510
of log x is 1/x.

00:33:31.510 --> 00:33:35.340
Between that and that, I've
just changed letters.

00:33:35.340 --> 00:33:40.820
That was like after the real
thinking of this lecture,

00:33:40.820 --> 00:33:45.530
which was the when x was an
input and y was an output, and

00:33:45.530 --> 00:33:48.610
I really needed two
different letters.

00:33:48.610 --> 00:33:50.120
OK, good, that's inverse
functions.

00:33:50.120 --> 00:33:52.226
Thank you.

00:33:52.226 --> 00:33:54.480
ANNOUNCER: This has been
a production of MIT

00:33:54.480 --> 00:33:56.870
OpenCourseWare and
Gilbert Strang.

00:33:56.870 --> 00:33:59.150
Funding for this video was
provided by the Lord

00:33:59.150 --> 00:34:00.360
Foundation.

00:34:00.360 --> 00:34:03.490
To help OCW continue to provide
free and open access

00:34:03.490 --> 00:34:06.570
to MIT courses please
make a donation at

00:34:06.570 --> 00:34:08.130
ocw.mit.edu/donate.