Labeling parts of a circle

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:05.380
Draw a circle and label the
radius, diameter, center,

00:00:05.380 --> 00:00:06.740
and the circumference.

00:00:06.740 --> 00:00:08.427
Let me draw a circle.

00:00:08.427 --> 00:00:10.260
And it won't be that
well drawn of a circle,

00:00:10.260 --> 00:00:11.468
but I think you get the idea.

00:00:11.468 --> 00:00:13.220
So that is my circle.

00:00:13.220 --> 00:00:16.300
I'm going to label
the center over here.

00:00:16.300 --> 00:00:17.210
I'll do the center.

00:00:17.210 --> 00:00:18.170
I'll call it c.

00:00:18.170 --> 00:00:19.850
So that is my center.

00:00:19.850 --> 00:00:21.240
And I'll draw an arrow there.

00:00:21.240 --> 00:00:23.530
That is the center
of the circle.

00:00:23.530 --> 00:00:26.660
And actually, the
circle itself is the set

00:00:26.660 --> 00:00:30.680
of all points that are a fixed
distance away from that center.

00:00:30.680 --> 00:00:32.630
And that fixed distance
away that they're

00:00:32.630 --> 00:00:35.460
all from that center,
that is the radius.

00:00:35.460 --> 00:00:37.690
So let me draw the radius.

00:00:37.690 --> 00:00:41.960
So this distance right
over here is the radius.

00:00:41.960 --> 00:00:44.190
That is the radius.

00:00:44.190 --> 00:00:47.550
And that's going to be the
same as this distance, which

00:00:47.550 --> 00:00:49.240
is the same as that distance.

00:00:49.240 --> 00:00:50.850
I can draw multiple radii.

00:00:50.850 --> 00:00:53.930
All of these are radii, the
distance between the center

00:00:53.930 --> 00:00:56.810
and any point on the circle.

00:00:56.810 --> 00:01:01.060
Now, a diameter just goes
straight across the circle,

00:01:01.060 --> 00:01:02.700
going through the center.

00:01:02.700 --> 00:01:04.819
From one side of the
circle to the other side,

00:01:04.819 --> 00:01:06.770
I'm going through the center.

00:01:06.770 --> 00:01:10.050
It's essentially two
radii put together.

00:01:10.050 --> 00:01:15.060
So for example, this
would be a diameter.

00:01:15.060 --> 00:01:18.702
You have one radii, than
another radii, all one line,

00:01:18.702 --> 00:01:20.660
going from one side of
the circle to the other,

00:01:20.660 --> 00:01:22.340
going through the center.

00:01:22.340 --> 00:01:24.070
So that is a diameter.

00:01:26.449 --> 00:01:27.990
And I could have
drawn it other ways.

00:01:27.990 --> 00:01:29.410
I could've drawn it like this.

00:01:29.410 --> 00:01:30.790
That would be another diameter.

00:01:30.790 --> 00:01:32.790
But they're going to have
the exact same length.

00:01:32.790 --> 00:01:35.340
And finally, we have to think
about the circumference.

00:01:35.340 --> 00:01:37.650
And the circumference
is really just how far

00:01:37.650 --> 00:01:39.780
you have to go to go
around the circle.

00:01:39.780 --> 00:01:41.880
Or if you put a
string on this circle,

00:01:41.880 --> 00:01:43.770
how long will that
string have to be?

00:01:43.770 --> 00:01:46.550
So what I'm tracing
out in blue right now,

00:01:46.550 --> 00:01:50.150
the length of what I'm tracing
out, is the circumference.

00:01:50.150 --> 00:01:54.790
So right over here, that
is the circumference.

00:01:59.040 --> 00:02:00.950
And we're done.


Solving logarithmic equations

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:03.760
We're asked to solve the
log of x plus log of 3

00:00:03.760 --> 00:00:06.840
is equal to 2 log
of 4 minus log of 2.

00:00:06.840 --> 00:00:08.530
So let me just rewrite it.

00:00:08.530 --> 00:00:15.080
So we have the log of x plus
the log of 3 is equal to 2 times

00:00:15.080 --> 00:00:21.150
the log of 4 minus the log
of 2, or the logarithm of 2.

00:00:21.150 --> 00:00:22.110
And this is a reminder.

00:00:22.110 --> 00:00:25.010
Whenever you see a logarithm
written without a base,

00:00:25.010 --> 00:00:26.560
the implicit base is 10.

00:00:26.560 --> 00:00:31.640
So we could write 10 here, 10
here, 10 here, and 10 here.

00:00:31.640 --> 00:00:33.130
But for the rest
of this example,

00:00:33.130 --> 00:00:34.620
I'll just skip writing
the 10 just because it'll

00:00:34.620 --> 00:00:35.730
save a little bit of time.

00:00:35.730 --> 00:00:38.320
But remember, this
literally means log base 10.

00:00:38.320 --> 00:00:39.820
So this expression,
right over here,

00:00:39.820 --> 00:00:42.580
is the power I have
to raise 10 to to get

00:00:42.580 --> 00:00:45.595
x, the power I have to
raise 10 to to get 3.

00:00:45.595 --> 00:00:47.220
Now with that out of
the way, let's see

00:00:47.220 --> 00:00:49.830
what logarithm
properties we can use.

00:00:49.830 --> 00:00:52.850
So we know, if we-- and
these are all the same base--

00:00:52.850 --> 00:01:00.790
we know that if we have log base
a of b plus log base a of c,

00:01:00.790 --> 00:01:06.110
then this is the same
thing as log base a of bc.

00:01:06.110 --> 00:01:08.574
And we also know-- so let
me write all the logarithm

00:01:08.574 --> 00:01:09.990
properties that
we know over here.

00:01:09.990 --> 00:01:15.834
We also know that if
we have a logarithm--

00:01:15.834 --> 00:01:17.750
let me write it this
way, actually-- if I have

00:01:17.750 --> 00:01:21.580
b times the log
base a of c, this

00:01:21.580 --> 00:01:26.847
is equal to log base a
of c to the bth power.

00:01:26.847 --> 00:01:28.680
And we also know, and
this is derived really

00:01:28.680 --> 00:01:31.490
straight from both of
these, is that if I

00:01:31.490 --> 00:01:37.170
have log base a of b
minus log base a of c,

00:01:37.170 --> 00:01:41.450
that this is equal to the
log base a of b over c.

00:01:41.450 --> 00:01:43.040
And this is really
straight derived

00:01:43.040 --> 00:01:45.850
from these two right over here.

00:01:45.850 --> 00:01:48.550
Now with that out of the way,
let's see what we can apply.

00:01:48.550 --> 00:01:51.590
So right over here, we have
all the logs are the same base.

00:01:51.590 --> 00:01:54.570
And we have logarithm of
x plus logarithm of 3.

00:01:54.570 --> 00:01:56.200
So by this property
right over here,

00:01:56.200 --> 00:01:58.810
the sum of logarithms
with the same base,

00:01:58.810 --> 00:02:02.840
this is going to be equal
to log base 3-- sorry,

00:02:02.840 --> 00:02:06.200
log base 10-- so I'll
just write it here.

00:02:06.200 --> 00:02:10.759
log base 10 of 3 times x, of 3x.

00:02:10.759 --> 00:02:14.320
Then, based on this
property right over here,

00:02:14.320 --> 00:02:16.700
this thing could be
rewritten-- so this

00:02:16.700 --> 00:02:19.200
is going to be equal to--
this thing can be written

00:02:19.200 --> 00:02:29.040
as log base 10 of 4 to the
second power, which is really

00:02:29.040 --> 00:02:30.940
just 16.

00:02:30.940 --> 00:02:33.680
So this is just going to be 16.

00:02:33.680 --> 00:02:39.680
And then we still have minus
logarithm base 10 of 2.

00:02:39.680 --> 00:02:42.300
And now, using
this last property,

00:02:42.300 --> 00:02:45.260
we know we have one logarithm
minus another logarithm.

00:02:45.260 --> 00:02:49.310
This is going to be equal
to log base 10 of 16

00:02:49.310 --> 00:02:55.880
over 2, 16 divided by 2,
which is the same thing as 8.

00:02:55.880 --> 00:02:59.060
So the right-hand side
simplifies to log base 10 of 8.

00:02:59.060 --> 00:03:04.570
The left-hand side
is log base 10 of 3x.

00:03:04.570 --> 00:03:08.430
So if 10 to some power is
going to be equal to 3x.

00:03:08.430 --> 00:03:11.210
And 10 to the same power
is going to be equal to 8.

00:03:11.210 --> 00:03:13.193
So 3x must be equal to 8.

00:03:16.760 --> 00:03:23.900
3x is equal to 8, and then we
can divide both sides by 3.

00:03:23.900 --> 00:03:29.770
Divide both sides by 3, you
get x is equal to 8 over 3.

00:03:29.770 --> 00:03:33.190
One way, this little step here,
I said, look, 10 to the-- this

00:03:33.190 --> 00:03:34.180
is an exponent.

00:03:34.180 --> 00:03:37.720
If I raise 10 to this exponent,
I get 3x, 10 to this exponent,

00:03:37.720 --> 00:03:38.600
I get 8.

00:03:38.600 --> 00:03:41.102
So 8 and 3x must
be the same thing.

00:03:41.102 --> 00:03:43.060
One other way you could
have thought about this

00:03:43.060 --> 00:03:47.170
is, let's take 10 to this
power, on both sides.

00:03:47.170 --> 00:03:51.570
So you could say 10 to
this power, and then 10

00:03:51.570 --> 00:03:53.430
to this power over here.

00:03:53.430 --> 00:03:55.920
If I raise 10 to
the power that I

00:03:55.920 --> 00:03:57.760
need to raise 10
to to get to 3x,

00:03:57.760 --> 00:04:00.020
well, I'm just going to get 3x.

00:04:00.020 --> 00:04:04.070
If I raise 10 to the power that
I need to raise 10 to to get 8,

00:04:04.070 --> 00:04:05.160
I'm just going to get 8.

00:04:05.160 --> 00:04:07.500
So once again, you've
got the 3x is equal to 8,

00:04:07.500 --> 00:04:08.970
and then you can simplify.

00:04:08.970 --> 00:04:12.020
You get x is equal to 8/3.


Calculator for natural logarithms

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:04.019
Find the natural log of
3 using a calculator.

00:00:04.019 --> 00:00:06.030
Round your answer to
the nearest hundredth.

00:00:06.030 --> 00:00:08.700
And just as a reminder,
natural log of 3,

00:00:08.700 --> 00:00:12.930
this is just another way
of saying log base e of 3,

00:00:12.930 --> 00:00:17.420
or what power do I have
to raise e to to get to 3?

00:00:17.420 --> 00:00:20.142
And 3 is a little bit
larger than e. e is 2.71

00:00:20.142 --> 00:00:21.350
and it goes on and on and on.

00:00:21.350 --> 00:00:23.490
So 3 is a little
bit larger than e

00:00:23.490 --> 00:00:26.610
so I'm guessing it's going to be
one point something something.

00:00:26.610 --> 00:00:29.420
So let's get our calculator
out and just calculate it.

00:00:29.420 --> 00:00:31.640
So natural log-- that's this
button right over here--

00:00:31.640 --> 00:00:35.030
ln-- I'm assuming it's
maybe from French log

00:00:35.030 --> 00:00:39.870
natural or something like
that-- So natural log of 3

00:00:39.870 --> 00:00:44.040
is one point-- and they want
us to round to the nearest

00:00:44.040 --> 00:00:45.220
hundredth.

00:00:45.220 --> 00:00:48.320
So it's 1.09, but
the thousandths place

00:00:48.320 --> 00:00:51.620
is an 8, which is greater
than or equal to 5

00:00:51.620 --> 00:00:53.910
so we want to round up.

00:00:53.910 --> 00:00:56.570
So we want to round the 9 up,
but that essentially takes us

00:00:56.570 --> 00:00:57.240
to 10.

00:00:57.240 --> 00:01:00.000
So 1.10.

00:01:00.000 --> 00:01:06.420
So this is approximately
equal to 1.10 when we round

00:01:06.420 --> 00:01:08.323
to the nearest hundredth.


Graphing natural logarithm function

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:04.481
We're asked to graph f(x) is equal to the natural log of

00:00:04.481 --> 00:00:08.545
2x, and I'll use a calculator here to find a table of values

00:00:08.545 --> 00:00:12.817
for different x values, what does f(x) equals, but then I'll hand draw the graph

00:00:12.817 --> 00:00:14.373
which might be a little bit ironic

00:00:14.373 --> 00:00:16.742
because I'll be using a graphic calculator to come up with the values.

00:00:16.742 --> 00:00:19.156
But I won't use the graphing function to graph it,

00:00:19.156 --> 00:00:21.316
we'll do that part by hand.

00:00:21.316 --> 00:00:23.452
So let's draw here a little table of values.

00:00:23.452 --> 00:00:25.310
Of x values and of y values.

00:00:25.310 --> 00:00:26.912
And see what we get.

00:00:26.912 --> 00:00:31.509
So these are my x values and let's just say y = f(x)

00:00:31.509 --> 00:00:33.878
So whatever the output is I'm going to set that equal

00:00:33.878 --> 00:00:36.920
to our dependent variable and plot it on the vertical axis

00:00:36.920 --> 00:00:38.870
and we call that dependent variable y.

00:00:38.870 --> 00:00:42.005
So let's just try really small numbers.

00:00:42.005 --> 00:00:44.327
So actually first, let's remind ourselves what's the domain?

00:00:44.327 --> 00:00:47.810
What's the set of valid inputs for x, that we can put in right there?

00:00:47.810 --> 00:00:51.595
So the natural logarithm is just logaritm base e.

00:00:51.595 --> 00:00:56.680
And any logarithm is only defined when input into the logaritm,

00:00:56.680 --> 00:01:00.372
in this case 2x, is greater than zero.

00:01:00.372 --> 00:01:02.462
You can't even take the logarithm of zero.

00:01:02.462 --> 00:01:04.621
You can raise something to a very negative exponent,

00:01:04.621 --> 00:01:08.151
negative bilion power, it will get pretty close to zero

00:01:08.151 --> 00:01:10.055
but you can never get to zero.

00:01:10.055 --> 00:01:13.352
If you take a positive base there is no exponent you can raise it to

00:01:13.352 --> 00:01:17.694
to get to the zero or get to a negative number.

00:01:17.694 --> 00:01:21.432
So the 2x here, the input into our logarithm function,

00:01:21.432 --> 00:01:23.545
in this case the natural logarithm function,

00:01:23.545 --> 00:01:25.937
it has to be greater than zero.

00:01:25.937 --> 00:01:28.584
And if that's greater than zero, divide both sides by 2.

00:01:28.584 --> 00:01:30.720
That means x is greater than zero.

00:01:30.720 --> 00:01:34.157
So that is essentially the constraint on our domain.

00:01:34.157 --> 00:01:38.453
Our domain is all real numbers greater than zero.

00:01:38.453 --> 00:01:41.843
So let's try something pretty close to zero.

00:01:41.843 --> 00:01:45.047
Just so we see what happens, the behaviour as we approach

00:01:45.047 --> 00:01:46.858
as we are close to zero.

00:01:46.858 --> 00:01:49.343
And especially as input here is less than one.

00:01:49.343 --> 00:02:00.813
So let's try 0.1, 0.5, 1.

00:02:00.813 --> 00:02:05.806
And let's try, I don't know - let's try 5...

00:02:05.806 --> 00:02:07.408
Oh, I don't actually want to get too far,

00:02:07.408 --> 00:02:10.310
cause I want to be able to see the resolution down here.

00:02:10.310 --> 00:02:15.976
So let's try 1.5 and let's try 3.

00:02:15.976 --> 00:02:19.157
So those will be our inputs that we'll try.

00:02:19.157 --> 00:02:21.967
That's sounds good enough.

00:02:21.967 --> 00:02:25.728
And then let me draw my axes and then we'll plot the points.

00:02:25.728 --> 00:02:29.002
So our domain is positive x values

00:02:29.002 --> 00:02:33.089
so we don't really have to draw much on the negative x values.

00:02:33.089 --> 00:02:35.411
But we will have some negative values here,

00:02:35.411 --> 00:02:38.337
so let me give ourselves some room to work with.

00:02:38.337 --> 00:02:40.287
And out x values go up to 3.

00:02:40.287 --> 00:02:44.885
So this is 1, 2 and 3.

00:02:44.885 --> 00:02:50.388
This is 0.5, 1.5 and this is 2.5 (which we do not use).

00:02:50.388 --> 00:02:54.869
And then let's see what our y values are, f(x) are going to be equal to

00:02:54.869 --> 00:02:56.890
So get out our TI-85

00:02:56.890 --> 00:03:00.048
So if we take the natural log -- remember, we have to take

00:03:00.048 --> 00:03:02.161
2 times x and then natural log of that.

00:03:02.161 --> 00:03:07.408
So if we take the natural log of 2 times 0.1,

00:03:07.408 --> 00:03:09.730
which is obviously 0.2, what do we get?

00:03:09.730 --> 00:03:14.281
We get negative 1.61, I'll say.

00:03:14.281 --> 00:03:17.462
Negative 1.61.

00:03:17.462 --> 00:03:25.682
And then if we input 0.5, what do we get?

00:03:25.682 --> 00:03:27.563
Let me get the calculator back.

00:03:27.563 --> 00:03:31.603
So we're going to take the natural log of 2 times 0.5.

00:03:31.603 --> 00:03:33.484
We can do it in out heads.

00:03:33.484 --> 00:03:35.388
Actually I'll just write it out just so it's clear what we're doing.

00:03:35.388 --> 00:03:39.429
2 times 0.5, that's the natural log of one.

00:03:39.429 --> 00:03:41.054
And you should be able to do that in your head.

00:03:41.054 --> 00:03:44.653
What power you have to raise any positive base to, to get to one?

00:03:44.653 --> 00:03:47.068
Well, you raise e to the zero power and you get one.

00:03:47.068 --> 00:03:50.203
So I'll write it over here.

00:03:50.203 --> 00:03:56.008
We should've been able do to that one in our heads.

00:03:56.008 --> 00:03:58.051
Now let's do the next one.

00:03:58.051 --> 00:04:07.432
What happens when we have the natural log of 2 times 1.

00:04:07.432 --> 00:04:09.313
Which is obviously just going to be 2.

00:04:09.313 --> 00:04:11.426
So it's te natural log of 2.

00:04:11.426 --> 00:04:14.746
Gets us .69. Which makes sense that this is less than 1.

00:04:14.746 --> 00:04:19.297
Because 2 is less than e. e is 2.71 and so on and so forth.

00:04:19.297 --> 00:04:21.596
So this is .69.

00:04:21.596 --> 00:04:30.071
Now let's try the natural log of 2 times 1.5.

00:04:30.071 --> 00:04:37.804
Natural log of 3.

00:04:37.804 --> 00:04:43.121
And that gets us to 1.10, I'll round to the hundedths.

00:04:43.121 --> 00:04:46.859
So this is 1.10.

00:04:46.859 --> 00:04:54.406
And finally the natural log of 2 times 3.

00:04:54.406 --> 00:04:58.818
2 times x, x is 3.

00:04:58.818 --> 00:05:01.720
What do I get? This is going to be natural log of 6.

00:05:01.720 --> 00:05:05.017
Which is 1.79.

00:05:05.017 --> 00:05:08.013
This is going to be, I'll choose a new color.

00:05:08.013 --> 00:05:13.051
This is going to be 1.79.

00:05:13.051 --> 00:05:19.228
So in terms of the coordinates we run as low as -1.61 and as high as 1.79.

00:05:19.228 --> 00:05:26.217
So let's call this right over here -1

00:05:26.217 --> 00:05:31.209
and then down here would be -2.

00:05:31.209 --> 00:05:34.971
I'll extend the y axis down a little bit.

00:05:34.971 --> 00:05:39.383
So this is the x axis and this is out y = f(x) axis.

00:05:39.383 --> 00:05:43.028
And then let's call this right over here positive 1.

00:05:43.028 --> 00:05:45.838
And this over here is positive 2.

00:05:45.838 --> 00:05:47.881
And this would be half way between those

00:05:47.881 --> 00:05:51.550
just cause it looks like we gonna have to be able to see that as well.

00:05:51.550 --> 00:05:55.614
And so this first point is 0.1, -1.61.

00:05:55.614 --> 00:05:58.145
So 0.1, that's one tenth,

00:05:58.145 --> 00:06:00.443
that's gonna be right around there.

00:06:00.443 --> 00:06:02.603
And then we have -1.61.

00:06:02.603 --> 00:06:06.875
It's gonna sit right about there.

00:06:06.875 --> 00:06:19.089
So that is the point (0.1, -1.61). Fair enough.

00:06:19.089 --> 00:06:22.108
That's the first point right there.

00:06:22.108 --> 00:06:24.848
Now let's do this one.

00:06:24.848 --> 00:06:26.821
0.5, 0

00:06:26.821 --> 00:06:29.817
When x is 0.5, y is 0.

00:06:29.817 --> 00:06:33.718
That's (0.5, 0). Fair enough.

00:06:33.718 --> 00:06:42.866
When x is equal to 1, y is 0.69.

00:06:42.866 --> 00:06:44.886
Which might be right about there.

00:06:44.886 --> 00:06:48.880
Just approximating it, so a little bit closer to 1 that is close 0.5.

00:06:48.880 --> 00:06:51.272
Actually a little bit closer to 0.5 than it is to 1.

00:06:51.272 --> 00:06:54.290
So let me put it right over there.

00:06:54.290 --> 00:06:59.608
This would be (1, 0.69)

00:06:59.608 --> 00:07:06.249
And then we have the point, when x is 1.5 - f(x) is 1.10.

00:07:06.249 --> 00:07:13.586
When x is 1.5, f(x) is 1.1, which takes us right above there.

00:07:13.586 --> 00:07:18.323
So that is the point and I think you see where this curve is going.

00:07:18.323 --> 00:07:21.736
(1.5, 1.10)

00:07:21.736 --> 00:07:24.221
And finally, so that was that point.

00:07:24.221 --> 00:07:26.009
We'll do this last one in yellow

00:07:26.009 --> 00:07:28.493
when x is 3, y is 1.79.

00:07:28.493 --> 00:07:33.811
So a little bit closer to 2.0 than to 1.5.

00:07:33.811 --> 00:07:36.528
So it's gonna be right about there.

00:07:36.528 --> 00:07:40.823
It's gonna be coordinate (3, 1.79).

00:07:40.823 --> 00:07:42.750
And now we can connect the dots

00:07:42.750 --> 00:07:44.538
and I'll do that in white.

00:07:44.538 --> 00:07:48.672
And so as we get x values that are closer and closer to 0

00:07:48.672 --> 00:07:52.619
Our graph of our function is gonna get more and more negative

00:07:52.619 --> 00:07:57.170
And it's gonna get closer and closer to the y axis without ever touching it.

00:07:57.170 --> 00:08:01.535
So it's gonna get less close from the y axis, slowly break away

00:08:01.535 --> 00:08:05.738
and then curve out like this.

00:08:05.738 --> 00:08:10.080
Curve out just like that.

00:08:10.080 --> 00:08:12.681
And just keep on going down like this.

00:08:12.681 --> 00:08:16.164
And what happens here is pretty cool, as x gets smaller and smaller

00:08:16.164 --> 00:08:21.388
The functions becomes more...infinetely negative as x aproaches 0.

00:08:21.388 --> 00:08:24.477
But x can never be 0.

00:08:24.477 --> 00:08:27.658
There is no power you can raise e or any positive base to,

00:08:27.658 --> 00:08:32.882
to actually get zero. You can raise it to raise to a very large negative exponent

00:08:32.882 --> 00:08:40.475
I you raise e to the negative one billion, you'll get a number

00:08:40.475 --> 00:08:43.540
That's very close to 0, because it is the same as

00:08:43.540 --> 00:08:45.886
1 over e to the one bilionth power.

00:08:45.886 --> 00:08:49.578
So this is a number that's very close to 0.

00:08:49.578 --> 00:08:51.598
But you're never going to approach zero.

00:08:51.598 --> 00:08:53.548
You can make this number more and more negative

00:08:53.548 --> 00:08:55.359
it'll just get smaller and smaller numbers

00:08:55.359 --> 00:08:59.074
but you'll never quite approach 0.

00:08:59.074 --> 00:09:03.788
So you can never a logarithm of zero. You just approach it.

00:09:03.788 --> 00:09:07.666
We're done! This is the graph of natural log of 2x.

00:09:07.666 --> 00:09:10.336
And it has a typical shape, cause it's just a logarithm.

00:09:10.336 --> 99:59:59.999
It has a shape of a logarithmic graph.


Natural logarithm with a calculator

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:06.030
Use a calculator to
find log base e of 67

00:00:06.030 --> 00:00:08.170
to the nearest thousandth.

00:00:08.170 --> 00:00:11.440
So just as a reminder, e is
one of these crazy numbers

00:00:11.440 --> 00:00:14.640
that shows up in nature, in
finance, and all these things,

00:00:14.640 --> 00:00:17.900
and it's approximately
equal to 2.71

00:00:17.900 --> 00:00:19.670
and it just keeps
going on and on and on.

00:00:19.670 --> 00:00:24.150
So you could view
log base e as 67.

00:00:24.150 --> 00:00:26.400
Let's see, what does e
mean? e is just a number,

00:00:26.400 --> 00:00:27.800
just like pi is just a number.

00:00:27.800 --> 00:00:31.289
So this is really the same
thing as saying log base 2.71,

00:00:31.289 --> 00:00:32.830
and the actual
numbers, so you'd have

00:00:32.830 --> 00:00:35.290
to write all the digits
that keep on going forever

00:00:35.290 --> 00:00:38.220
and never repeat 6 of 67.

00:00:38.220 --> 00:00:42.240
So what power do I have to
raise e to to get to 67?

00:00:42.240 --> 00:00:45.170
So another way of saying
that is this is equal to x.

00:00:45.170 --> 00:00:49.340
You're saying e to
the x is equal to 67,

00:00:49.340 --> 00:00:51.720
we need to figure out what x is.

00:00:51.720 --> 00:00:53.620
Now, traditionally
you will never

00:00:53.620 --> 00:00:57.960
see someone write log
base e even though e

00:00:57.960 --> 00:01:01.340
is one of the most common
bases to take a logarithm of.

00:01:01.340 --> 00:01:04.030
And so the reason why you
wouldn't see log base e written

00:01:04.030 --> 00:01:06.220
this way is log
base e is referred

00:01:06.220 --> 00:01:08.480
to as the natural logarithm.

00:01:08.480 --> 00:01:11.640
And I think that's used because
e shows up so many times

00:01:11.640 --> 00:01:12.680
in nature.

00:01:12.680 --> 00:01:16.464
So log base e of 67, another
way of saying that-- or seeing

00:01:16.464 --> 00:01:18.130
that, and the more
typical way of seeing

00:01:18.130 --> 00:01:20.800
that is the natural log.

00:01:20.800 --> 00:01:22.780
And I think this
is ln, so I think

00:01:22.780 --> 00:01:27.720
it's maybe from French or
something, log natural, of 67.

00:01:27.720 --> 00:01:31.530
So this is the same thing
as log base e of 67.

00:01:31.530 --> 00:01:33.320
This is saying the
exact same thing.

00:01:33.320 --> 00:01:36.230
To what power do I have
to raise e to to get 67?

00:01:36.230 --> 00:01:40.284
When you see this ln, it
literally means log base e.

00:01:40.284 --> 00:01:41.700
Now, they let us
use a calculator,

00:01:41.700 --> 00:01:44.660
and that's good because I don't
know off the top of my head

00:01:44.660 --> 00:01:48.630
what power I have to raise
2.71 so on and so forth-- what

00:01:48.630 --> 00:01:50.810
power I have to raise
that to to get to 67.

00:01:50.810 --> 00:01:53.050
So we'll get our calculator out.

00:01:53.050 --> 00:01:55.882
So we get the TI85 out.

00:01:55.882 --> 00:01:57.340
And different
calculators will have

00:01:57.340 --> 00:01:58.500
different ways of doing it.

00:01:58.500 --> 00:02:00.620
If you have a graphing
calculator like this,

00:02:00.620 --> 00:02:03.710
you literally can literally type
in the statement natural log

00:02:03.710 --> 00:02:05.600
of 67 then evaluate it.

00:02:05.600 --> 00:02:07.670
So here this is
the button for ln,

00:02:07.670 --> 00:02:10.830
means natural log,
log natural, maybe.

00:02:10.830 --> 00:02:13.951
ln of 67, and then
you press Enter,

00:02:13.951 --> 00:02:15.200
and it'll give you the answer.

00:02:15.200 --> 00:02:16.866
If you don't have a
graphing calculator,

00:02:16.866 --> 00:02:20.749
you might have to press 67
and then press natural log

00:02:20.749 --> 00:02:22.790
to give you the answer,
but a graphing calculator

00:02:22.790 --> 00:02:25.331
can literally type it in the
way that you would write it out,

00:02:25.331 --> 00:02:26.630
and then you would press Enter.

00:02:26.630 --> 00:02:31.780
So 4.20469 and we want to round
to the nearest thousandth.

00:02:31.780 --> 00:02:34.140
So this is the thousandths
place right here, this 4.

00:02:34.140 --> 00:02:37.090
The digit after that is
5 or larger, it's a 6,

00:02:37.090 --> 00:02:38.310
so we're going to round up.

00:02:38.310 --> 00:02:41.680
So this is 4.205.

00:02:41.680 --> 00:02:49.589
So this is approximately
equal to 4.205.

00:02:49.589 --> 00:02:51.130
And it actually
makes a lot of sense,

00:02:51.130 --> 00:02:54.250
because we know that
e is greater than 2,

00:02:54.250 --> 00:02:58.430
and it is less than 3.

00:02:58.430 --> 00:03:02.320
And if you think about
what 2 to the fourth power

00:03:02.320 --> 00:03:04.270
gets you to 16.

00:03:04.270 --> 00:03:09.040
And 3 to the fourth
power gets you to 81.

00:03:09.040 --> 00:03:14.270
67 is between 16 and 81
and e is between 2 and 3.

00:03:14.270 --> 00:03:16.540
So at least it
feels right that's

00:03:16.540 --> 00:03:19.120
something that's like
2.71 to the little

00:03:19.120 --> 00:03:22.170
over the fourth power should
get you to a number that's

00:03:22.170 --> 00:03:23.872
pretty close to 3
to the fourth power.

00:03:23.872 --> 00:03:25.830
Actually that makes sense
because it's actually

00:03:25.830 --> 00:03:27.300
closer to 3.

00:03:27.300 --> 00:03:30.380
2.71 is closer to
3 than it is to 2.

00:03:30.380 --> 00:03:32.780
So this feels right, that
you take this to the fourth,

00:03:32.780 --> 00:03:36.320
little over the fourth
power, you get to 67.


Using multiple logarithm properties to simplify

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:07.730
We're asked to simplify log base
5 of 25 to the x power over y.

00:00:07.730 --> 00:00:09.640
So we can use some
logarithm properties.

00:00:09.640 --> 00:00:12.790
And I do agree that this does
require some simplification

00:00:12.790 --> 00:00:14.840
over here, that having
this right over here

00:00:14.840 --> 00:00:18.080
inside of the logarithm is not
a pleasant thing to look at.

00:00:18.080 --> 00:00:20.150
So the first thing that
we realize-- and this

00:00:20.150 --> 00:00:23.110
is one of our
logarithm properties--

00:00:23.110 --> 00:00:25.360
is logarithm for a given
base-- so let's say

00:00:25.360 --> 00:00:28.640
that the base is
x-- of a/b, that

00:00:28.640 --> 00:00:34.750
is equal to log base x of
a minus log base x of b.

00:00:34.750 --> 00:00:37.740
And here we have
25 to the x over y.

00:00:37.740 --> 00:00:39.160
So we can simplify.

00:00:39.160 --> 00:00:40.310
So let me write this down.

00:00:40.310 --> 00:00:42.120
I'll do this in blue.

00:00:42.120 --> 00:00:48.770
Log base 5 of 25 to the x over
y using this property means

00:00:48.770 --> 00:00:52.730
that it's the same thing
as log base 5 of 25

00:00:52.730 --> 00:00:58.660
to the x power minus
log base 5 of y.

00:00:58.660 --> 00:01:02.340
Now, this looks like we can do
a little bit of simplifying.

00:01:02.340 --> 00:01:05.010
It seems like the relevant
logarithm property here

00:01:05.010 --> 00:01:10.430
is if I have log base x of
a to the b power, that's

00:01:10.430 --> 00:01:14.620
the same thing as
b times log base x

00:01:14.620 --> 00:01:16.700
of a, that this
exponent over here

00:01:16.700 --> 00:01:19.540
can be moved out front, which
is what we did it right over

00:01:19.540 --> 00:01:20.292
there.

00:01:20.292 --> 00:01:21.750
So this part right
over here can be

00:01:21.750 --> 00:01:29.420
rewritten as x times the
logarithm base 5 of 25.

00:01:29.420 --> 00:01:35.020
And then, of course, we
have minus log base 5 of y.

00:01:35.020 --> 00:01:38.810
And this is useful
because log base 5 of 25

00:01:38.810 --> 00:01:40.830
is actually fairly
easy to think about.

00:01:40.830 --> 00:01:42.650
This part right
here is asking us,

00:01:42.650 --> 00:01:47.260
what power do I have to
raise 5 to to get to 25?

00:01:47.260 --> 00:01:50.100
So we have to raise 5 to the
second power to get to 25.

00:01:50.100 --> 00:01:52.390
So this simplifies to 2.

00:01:52.390 --> 00:01:56.550
So then we are left
with, this is equal to--

00:01:56.550 --> 00:02:00.540
and I'll write it in front
of the x now-- 2 times

00:02:00.540 --> 00:02:06.950
x minus log base 5 of y.

00:02:06.950 --> 00:02:08.742
And we're done.


Logarithm of a power

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:04.859
We're asked to simplify log
base 5 of x to the third.

00:00:04.859 --> 00:00:06.650
And once again, we're
just going to rewrite

00:00:06.650 --> 00:00:08.010
this in a different way.

00:00:08.010 --> 00:00:11.020
You could argue whether it's
going to be more simple or not.

00:00:11.020 --> 00:00:13.740
And the logarithm
property that I'm

00:00:13.740 --> 00:00:17.230
guessing that we should use
for this example right here

00:00:17.230 --> 00:00:24.420
is the property-- if I take
log base x of-- let me pick

00:00:24.420 --> 00:00:29.600
some more letters here, log
base x of y to the zth power.

00:00:29.600 --> 00:00:36.860
This is the same thing as
z times log base x of y.

00:00:36.860 --> 00:00:38.380
So this is a logarithm property.

00:00:38.380 --> 00:00:41.080
If I'm taking the logarithm
of a given base of something

00:00:41.080 --> 00:00:46.040
to a power, I could take
that power out front

00:00:46.040 --> 00:00:48.570
and multiply that times
the log of the base,

00:00:48.570 --> 00:00:51.100
of just the y in this case.

00:00:51.100 --> 00:00:52.770
So we apply this
property over here.

00:00:52.770 --> 00:00:54.690
And in a second, once
I do this problem,

00:00:54.690 --> 00:00:57.610
we'll talk about why this
actually makes a lot of sense

00:00:57.610 --> 00:01:00.160
and comes straight out
of exponent properties.

00:01:00.160 --> 00:01:01.700
But if we just apply
that over here,

00:01:01.700 --> 00:01:05.309
we get log base 5
of x to the third.

00:01:05.309 --> 00:01:07.100
Well, this is the
exponent right over here.

00:01:07.100 --> 00:01:09.040
That's the same thing as z.

00:01:09.040 --> 00:01:10.610
So that's going to
be the same thing

00:01:10.610 --> 00:01:12.620
as-- let me do this
in a different color--

00:01:12.620 --> 00:01:15.161
that 3 is the same thing-- we
could put it out front-- that's

00:01:15.161 --> 00:01:21.315
the same thing as 3 times
the logarithm base 5 of x.

00:01:21.315 --> 00:01:21.940
And we're done.

00:01:21.940 --> 00:01:24.398
This is just another way of
writing it using this property.

00:01:24.398 --> 00:01:27.100
And so you could argue that
this is a what-- maybe this

00:01:27.100 --> 00:01:28.750
is a simplification
because you took

00:01:28.750 --> 00:01:30.940
the exponent outside
of the logarithm,

00:01:30.940 --> 00:01:34.122
and you're multiplying the
logarithm by that number now.

00:01:34.122 --> 00:01:35.830
Now with that out of
the way, let's think

00:01:35.830 --> 00:01:38.540
about why that
actually makes sense.

00:01:38.540 --> 00:01:43.700
So let's say that we
know that-- and I'll just

00:01:43.700 --> 00:01:45.210
pick some arbitrary
letters here--

00:01:45.210 --> 00:01:50.456
let's say that we know that a
to the b power is equal to c.

00:01:50.456 --> 00:01:52.080
And so if we know
that-- that's written

00:01:52.080 --> 00:01:53.500
as an exponential equation.

00:01:53.500 --> 00:01:55.450
If we wanted to
write the same truth

00:01:55.450 --> 00:02:02.290
as a logarithmic equation, we
would say logarithm base a of c

00:02:02.290 --> 00:02:03.750
is equal to b.

00:02:03.750 --> 00:02:06.280
To what power do I have
to raise a to get c?

00:02:06.280 --> 00:02:08.120
I raise it to the bth power.

00:02:08.120 --> 00:02:10.830
a to the b power is equal to c.

00:02:10.830 --> 00:02:12.060
Fair enough.

00:02:12.060 --> 00:02:14.950
Now let's take both sides of
this equation right over here,

00:02:14.950 --> 00:02:17.220
and raise it to the dth power.

00:02:17.220 --> 00:02:19.280
So let me make it--
so let's raise--

00:02:19.280 --> 00:02:22.150
take both sides of this equation
and raise it to the dth power.

00:02:22.150 --> 00:02:23.110
Instead of doing
it in place, I'm

00:02:23.110 --> 00:02:24.670
just going to
rewrite it over here.

00:02:24.670 --> 00:02:26.503
So I wrote the original
equation, a to the b

00:02:26.503 --> 00:02:29.490
is equal to c, which is just
rewriting this statement.

00:02:29.490 --> 00:02:33.540
But let me take both sides
of this to the dth power.

00:02:33.540 --> 00:02:34.810
And I should be consistent.

00:02:34.810 --> 00:02:36.400
I'll use all capital letters.

00:02:36.400 --> 00:02:38.780
So this should be a
B. Actually, let's

00:02:38.780 --> 00:02:40.480
say I'm using all
lowercase letters.

00:02:40.480 --> 00:02:42.010
This is a lowercase c.

00:02:42.010 --> 00:02:45.300
So let me write it
this way, a to the-- so

00:02:45.300 --> 00:02:48.536
I'm going to raise
this to the dth power,

00:02:48.536 --> 00:02:50.410
and I'm going to raise
this to the dth power.

00:02:50.410 --> 00:02:52.390
Obviously, if these two things
are equal to each other,

00:02:52.390 --> 00:02:54.360
if I raise both sides
to the same power,

00:02:54.360 --> 00:02:57.520
the equality is
still going to hold.

00:02:57.520 --> 00:03:01.760
Now, what's
interesting over here

00:03:01.760 --> 00:03:04.360
is we can now say--
what we could do

00:03:04.360 --> 00:03:07.437
is we can use what we know
about exponent properties.

00:03:07.437 --> 00:03:09.020
Say, look, if I have
a to the b power,

00:03:09.020 --> 00:03:11.050
and then I raise
that to the d power,

00:03:11.050 --> 00:03:14.340
our exponent properties say
that this is the same thing.

00:03:14.340 --> 00:03:22.060
This is equal to
a to the bd power.

00:03:22.060 --> 00:03:23.965
This is equal to a to the bd.

00:03:23.965 --> 00:03:24.840
Let me write it here.

00:03:24.840 --> 00:03:26.830
This is-- let me do that
in a different color.

00:03:26.830 --> 00:03:28.720
I've already used that green.

00:03:28.720 --> 00:03:31.270
This right over
here, using what we

00:03:31.270 --> 00:03:32.770
know about exponent
properties, this

00:03:32.770 --> 00:03:35.640
is the same thing as
a to the bd power.

00:03:35.640 --> 00:03:40.660
So we have a to the bd power
is equal to c to the dth power.

00:03:40.660 --> 00:03:42.995
And now this
exponential equation,

00:03:42.995 --> 00:03:45.900
if we would write it as
a logarithmic equation,

00:03:45.900 --> 00:03:58.810
we would say log base a of c to
the dth power is equal to bd.

00:03:58.810 --> 00:04:02.760
What power do I have to raise
a to get to c to the dth power?

00:04:02.760 --> 00:04:03.680
To get to this?

00:04:03.680 --> 00:04:05.810
I have to raise it
to the bd power.

00:04:05.810 --> 00:04:07.620
But what do we know that b is?

00:04:07.620 --> 00:04:12.000
We already know that b is
this thing right over here.

00:04:12.000 --> 00:04:15.020
So if we substitute
this in for b,

00:04:15.020 --> 00:04:19.180
and we can rewrite
this as db, we

00:04:19.180 --> 00:04:25.670
get logarithm base a of c to
the dth power is equal to bd,

00:04:25.670 --> 00:04:29.130
or you could also call that
db, if you switch the order.

00:04:29.130 --> 00:04:32.476
And so that's
equal to d times b.

00:04:32.476 --> 00:04:38.590
b is just log base a of c.

00:04:38.590 --> 00:04:39.500
So there you have it.

00:04:39.500 --> 00:04:40.830
We just derived this property.

00:04:40.830 --> 00:04:42.840
Log base a of c
to the dth, that's

00:04:42.840 --> 00:04:45.300
the same thing as
d times log base a

00:04:45.300 --> 00:04:48.860
of c, which we applied
right over here.


Sum of logarithms with same base

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:04.650
We're asked to simplify
log base 3 of 27x.

00:00:04.650 --> 00:00:06.400
And frankly, this is
already quite simple.

00:00:06.400 --> 00:00:09.620
But I'm assuming they want us
to use some logarithm properties

00:00:09.620 --> 00:00:11.830
and manipulate this in
some way, maybe to actually

00:00:11.830 --> 00:00:13.550
make it a little bit
more complicated.

00:00:13.550 --> 00:00:15.260
But let's give our
best shot at it.

00:00:15.260 --> 00:00:17.740
So the logarithm property
that jumps out at me--

00:00:17.740 --> 00:00:20.440
because this right over here--
we're saying what power do we

00:00:20.440 --> 00:00:23.050
have to raise 3 to to get 27x?

00:00:23.050 --> 00:00:26.350
27x is the same
thing as 27 times x.

00:00:26.350 --> 00:00:30.210
And so the logarithm property it
seems like they want us to use

00:00:30.210 --> 00:00:36.670
is log base-- let me write
it-- log base b of a times

00:00:36.670 --> 00:00:40.280
c-- I'll write it this way--
log base b of a times c.

00:00:40.280 --> 00:00:44.850
This is equal to the
logarithm base b of a plus

00:00:44.850 --> 00:00:48.430
the logarithm base b of c.

00:00:48.430 --> 00:00:51.090
And this comes straight out
of the exponent properties

00:00:51.090 --> 00:00:54.757
that if you have two exponents,
two with the same base,

00:00:54.757 --> 00:00:55.840
you can add the exponents.

00:00:55.840 --> 00:00:57.620
So let me make that a
little bit clearer to you.

00:00:57.620 --> 00:00:58.850
And if this part is
a little confusing,

00:00:58.850 --> 00:01:00.120
the important part
for this example

00:01:00.120 --> 00:01:01.750
is that you know
how to apply this.

00:01:01.750 --> 00:01:04.269
But it's even better if
you know the intuition.

00:01:04.269 --> 00:01:10.430
So let's say that log base b
of a times c is equal to x.

00:01:10.430 --> 00:01:13.680
So this thing right over
here evaluates to x.

00:01:13.680 --> 00:01:17.340
Let's say that this thing
right over here evaluates to y.

00:01:17.340 --> 00:01:22.070
So log base b of
a is equal to y.

00:01:22.070 --> 00:01:26.580
And let's say that this thing
over here evaluates to z.

00:01:26.580 --> 00:01:32.820
So log base b of
c is equal to z.

00:01:32.820 --> 00:01:37.860
Now, what we know is, this thing
right over here or this thing

00:01:37.860 --> 00:01:43.290
right over here tells
us that b to the x power

00:01:43.290 --> 00:01:46.950
is equal to a times c.

00:01:46.950 --> 00:01:49.550
Now, this right over
here is telling us

00:01:49.550 --> 00:01:51.743
that b to the y
power is equal to a.

00:01:54.620 --> 00:01:58.240
And this over here is telling
us that b to the z power

00:01:58.240 --> 00:01:59.670
is equal to c.

00:01:59.670 --> 00:02:01.610
Let me do that in
that same green.

00:02:01.610 --> 00:02:03.840
So I'm just writing
the same truth.

00:02:03.840 --> 00:02:05.810
I'm writing it as an
exponential function

00:02:05.810 --> 00:02:09.210
or exponential equation, instead
of a logarithmic equation.

00:02:09.210 --> 00:02:12.800
So b to the zth
power is equal to c.

00:02:12.800 --> 00:02:18.370
This is the same statement
or the same truth

00:02:18.370 --> 00:02:19.860
said in a different way.

00:02:19.860 --> 00:02:23.190
And this is the same truth
said in a different way.

00:02:23.190 --> 00:02:26.590
Well, if we know that
a is equal to this,

00:02:26.590 --> 00:02:34.030
is equal to b to the y,
and c is equal to bz,

00:02:34.030 --> 00:02:37.400
then we can write
b to the x power

00:02:37.400 --> 00:02:42.540
is equal to b to the y
power-- that's what a is.

00:02:42.540 --> 00:02:46.275
We know that already--
times b to the z power.

00:02:49.390 --> 00:02:53.830
And we know from our
exponent properties

00:02:53.830 --> 00:02:56.650
that if we take b to
the y times b to the z,

00:02:56.650 --> 00:03:00.340
this is the same thing
as b to the-- I'll

00:03:00.340 --> 00:03:04.260
do it in a neutral color--
b to the y plus z power.

00:03:04.260 --> 00:03:06.770
This comes straight out of
our exponent properties.

00:03:06.770 --> 00:03:08.764
And so if b to
the y plus z power

00:03:08.764 --> 00:03:10.930
is the same thing as b to
the x power, that tells us

00:03:10.930 --> 00:03:14.650
that x must be
equal to y plus z.

00:03:18.720 --> 00:03:21.370
If this is confusing to you,
don't worry about it too much.

00:03:21.370 --> 00:03:23.490
The important thing, or at
least the first important thing,

00:03:23.490 --> 00:03:24.920
is that you know
how to apply it.

00:03:24.920 --> 00:03:26.560
And then you can think about
this a little bit more,

00:03:26.560 --> 00:03:28.476
and you can even try it
out with some numbers.

00:03:28.476 --> 00:03:30.610
You just have to realize
that logarithms are really

00:03:30.610 --> 00:03:31.510
just exponents.

00:03:31.510 --> 00:03:33.560
I know when people first would
tell me that, I was like, well,

00:03:33.560 --> 00:03:34.393
what does that mean?

00:03:34.393 --> 00:03:36.782
But when you
evaluate a logarithm,

00:03:36.782 --> 00:03:38.240
you're getting an
exponent that you

00:03:38.240 --> 00:03:41.370
would have to raise b
to to get to a times c.

00:03:41.370 --> 00:03:44.350
But let's just apply this
property right over here.

00:03:44.350 --> 00:03:45.870
So if we apply it
to this one, we

00:03:45.870 --> 00:03:50.670
know that log base
3 of 27 times x--

00:03:50.670 --> 00:03:56.460
I'll write it that way-- is
equal to log base 3 of 27

00:03:56.460 --> 00:04:02.660
plus log base 3 of x.

00:04:02.660 --> 00:04:05.510
And then this right over
here, we can evaluate.

00:04:05.510 --> 00:04:07.690
This tells us, what
power do I have

00:04:07.690 --> 00:04:10.420
to raise 3 to to get to 27?

00:04:10.420 --> 00:04:11.710
You could view it as this way.

00:04:11.710 --> 00:04:13.060
3 to the ?

00:04:13.060 --> 00:04:15.500
is equal to 27.

00:04:15.500 --> 00:04:18.350
Well, 3 to the third
power is equal to 27.

00:04:18.350 --> 00:04:20.870
3 times 3 is 9 times 3 is 27.

00:04:20.870 --> 00:04:23.367
So this right over
here evaluates to 3.

00:04:23.367 --> 00:04:24.950
So if we were to
simplify-- or I guess

00:04:24.950 --> 00:04:26.320
I wouldn't even call
it simplifying it.

00:04:26.320 --> 00:04:28.861
I would just call it expanding
it out or using this property,

00:04:28.861 --> 00:04:31.150
because we now have two
terms where we started off

00:04:31.150 --> 00:04:31.734
with one term.

00:04:31.734 --> 00:04:33.150
Actually, if we
started with this,

00:04:33.150 --> 00:04:35.530
I'd say that this is the
more simple version of it.

00:04:35.530 --> 00:04:38.590
But when we rewrite it,
this first term becomes 3.

00:04:41.560 --> 00:04:46.060
And then we're left with
plus log base 3 of x.

00:04:46.060 --> 00:04:48.480
So this is just an
alternate way of writing

00:04:48.480 --> 00:04:54.220
this original statement,
log base 3 of 27x.

00:04:54.220 --> 00:04:55.830
So once again, not
clear that this

00:04:55.830 --> 00:04:58.240
is simpler than this
right over here.

00:04:58.240 --> 00:05:02.798
It's just another way of writing
it using logarithm properties.


Graphing logarithmic functions

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:05.690
We're asked to graph, y is
equal to log base 5 of x.

00:00:05.690 --> 00:00:07.780
And just to remind us
what this is saying,

00:00:07.780 --> 00:00:10.240
this is saying that y
is equal to the power

00:00:10.240 --> 00:00:13.460
that I have to raise
5 to to get to x.

00:00:13.460 --> 00:00:15.980
Or if I were to write
this logarithmic equation

00:00:15.980 --> 00:00:19.510
as an exponential
equation, 5 is my base,

00:00:19.510 --> 00:00:23.640
y is the exponent that I
have to raise my base to,

00:00:23.640 --> 00:00:28.420
and then x is what I get when
I raise 5 to the yth power.

00:00:28.420 --> 00:00:32.570
So another way of writing
this equation would be 5

00:00:32.570 --> 00:00:40.710
to the y'th power is
going to be equal to x.

00:00:40.710 --> 00:00:43.010
These are the same thing.

00:00:43.010 --> 00:00:45.480
Here, we have y as
a function of x.

00:00:45.480 --> 00:00:48.550
Here, we have x as
a function of y.

00:00:48.550 --> 00:00:50.980
But they're really saying
the exact same thing,

00:00:50.980 --> 00:00:53.624
raise 5 to the y'th
power to get x.

00:00:53.624 --> 00:00:55.790
When you put it as a
logarithm, you're saying, well,

00:00:55.790 --> 00:00:58.010
what power do I have
to raise 5 to to get x?

00:00:58.010 --> 00:00:59.440
We'll have to raise it to y.

00:00:59.440 --> 00:01:02.940
Here, what do I get when I
raise five to the y power?

00:01:02.940 --> 00:01:04.099
I get x.

00:01:04.099 --> 00:01:07.200
That out of the way, let's
make ourselves a little table

00:01:07.200 --> 00:01:08.710
that we can use to
plot some points,

00:01:08.710 --> 00:01:10.335
and then we can
connect the dots to see

00:01:10.335 --> 00:01:11.980
what this curve looks like.

00:01:11.980 --> 00:01:13.940
So let me pick some
x's and some y's.

00:01:18.500 --> 00:01:21.100
And we, in general, want to
pick some numbers that give us

00:01:21.100 --> 00:01:24.689
some nice round answers, some
nice fairly simple numbers

00:01:24.689 --> 00:01:26.230
for us to deal with,
so that we don't

00:01:26.230 --> 00:01:27.760
have to get the calculator.

00:01:27.760 --> 00:01:29.590
And so in general,
you want to pick

00:01:29.590 --> 00:01:34.250
x values where the power
that you have to raise 5

00:01:34.250 --> 00:01:38.156
to to get that x value is a
pretty straightforward power.

00:01:38.156 --> 00:01:39.530
Or another way to
think about it,

00:01:39.530 --> 00:01:41.810
you could just think about
the different y values

00:01:41.810 --> 00:01:44.740
that you want to raise
5 to the power of,

00:01:44.740 --> 00:01:46.360
and then you could
get your x values.

00:01:46.360 --> 00:01:48.590
So we could actually
think about this one

00:01:48.590 --> 00:01:52.510
to come up with our
actual x values.

00:01:52.510 --> 00:01:56.370
But we want to be clear that
when we express it like this,

00:01:56.370 --> 00:01:59.850
the independent variable is x,
and the dependent variable is

00:01:59.850 --> 00:02:00.480
y.

00:02:00.480 --> 00:02:03.660
We might just look at
this one to pick some nice

00:02:03.660 --> 00:02:10.440
even or nice x's that give
us nice clean answers for y.

00:02:10.440 --> 00:02:12.690
So here, I'm actually going
to fill in the y first,

00:02:12.690 --> 00:02:14.820
just so we get nice clean x's.

00:02:14.820 --> 00:02:16.500
So let's say we're
going to raise five

00:02:16.500 --> 00:02:19.120
to the-- let's say we're going
to raise it-- I'm going to pick

00:02:19.120 --> 00:02:23.820
some new colors-- negative
2, negative 2 power--

00:02:23.820 --> 00:02:30.490
and let me do some other
colors-- negative 1, 0, 1.

00:02:30.490 --> 00:02:33.620
I'll do one more, and then 2.

00:02:33.620 --> 00:02:36.660
So once again, this is
a little nontraditional,

00:02:36.660 --> 00:02:38.890
where I'm filling in the
dependent variable first.

00:02:38.890 --> 00:02:40.300
But the way that
we've written it over

00:02:40.300 --> 00:02:42.290
here, it's actually given
the dependent variable,

00:02:42.290 --> 00:02:44.750
it's easy to figure out what
the independent variable needs

00:02:44.750 --> 00:02:47.130
to be for this
logarithmic function.

00:02:47.130 --> 00:02:50.420
So, what x gives me
a y of negative 2?

00:02:50.420 --> 00:02:52.640
What x gives me--
what does x have

00:02:52.640 --> 00:02:55.430
to be for y to be
equal to negative 2?

00:02:55.430 --> 00:02:59.590
Well, 5 to the negative 2 power
is going to be equal to x.

00:02:59.590 --> 00:03:04.480
So 5 to the negative
2 is 1 over 25.

00:03:04.480 --> 00:03:07.440
So we get 1 over 25.

00:03:07.440 --> 00:03:09.000
If we go back to
this earlier one,

00:03:09.000 --> 00:03:13.200
if we say log base
5 of 1 over 25,

00:03:13.200 --> 00:03:16.530
what power do I have to
raise 5 to to get 1 over 25?

00:03:16.530 --> 00:03:19.420
We'll have to raise it
to the negative 2 power.

00:03:19.420 --> 00:03:22.110
Or you could say 5
to the negative 2

00:03:22.110 --> 00:03:24.460
is equal to 1 over 25.

00:03:24.460 --> 00:03:27.960
These are saying the
exact same thing.

00:03:27.960 --> 00:03:30.160
Now let's do another one.

00:03:30.160 --> 00:03:34.100
What happens when I raise
5 to the negative 1 power?

00:03:34.100 --> 00:03:35.487
I get one fifth.

00:03:35.487 --> 00:03:37.320
So if we go to this
original one over there,

00:03:37.320 --> 00:03:42.670
we're just saying that
log base 5 of one fifth.

00:03:42.670 --> 00:03:44.210
Want to be careful.

00:03:44.210 --> 00:03:46.810
This is saying, what power
do I have to raise 5 to

00:03:46.810 --> 00:03:48.130
in order to get one fifth.

00:03:48.130 --> 00:03:50.640
We'll have to raise it
to the negative 1 power.

00:03:53.150 --> 00:03:55.080
What happens when I take
5 to the 0'th power?

00:03:55.080 --> 00:03:57.249
I get one.

00:03:57.249 --> 00:03:59.290
And so this relationship--
This is the same thing

00:03:59.290 --> 00:04:02.650
as saying log base 5 of 1.

00:04:02.650 --> 00:04:05.430
What power do I have
to raise 5 to to get 1?

00:04:05.430 --> 00:04:08.910
I just have to raise
it to the 0th power.

00:04:08.910 --> 00:04:10.660
Let's do the next two.

00:04:10.660 --> 00:04:13.470
What happens when I raise
5 to the first power?

00:04:13.470 --> 00:04:17.180
Well, I get 5 So if you go look
over here, that's just saying,

00:04:17.180 --> 00:04:20.410
log, what power do I have
to raise 5 to to get 5?

00:04:20.410 --> 00:04:23.880
We'll have to just raise
it to the first power.

00:04:23.880 --> 00:04:28.800
And then finally, if I
take 5 squared, I get 25.

00:04:28.800 --> 00:04:31.680
So when you look at it from
the logarithmic point of view,

00:04:31.680 --> 00:04:34.410
you say, well, what power
do I have to raise 5 to

00:04:34.410 --> 00:04:36.020
to get to 25?

00:04:36.020 --> 00:04:38.930
We'll have to raise it
to the second power.

00:04:38.930 --> 00:04:41.830
So I took the inverse of
the logarithmic function.

00:04:41.830 --> 00:04:43.550
I wrote it as an
exponential function.

00:04:43.550 --> 00:04:47.270
I switched the dependent
and independent variables,

00:04:47.270 --> 00:04:50.760
so I can derive nice clean x's
that will give me nice clean

00:04:50.760 --> 00:04:51.734
y's.

00:04:51.734 --> 00:04:54.150
Now with that out of the way,
but I do want to remind you,

00:04:54.150 --> 00:04:57.497
I could have just picked
random numbers over here,

00:04:57.497 --> 00:04:59.830
but then I would have probably
gotten less clean numbers

00:04:59.830 --> 00:05:00.260
over here.

00:05:00.260 --> 00:05:01.510
I would have had to
use a calculator.

00:05:01.510 --> 00:05:03.093
The only reason why
I did it this way,

00:05:03.093 --> 00:05:06.810
is so I get nice clean results
that I can plot by hand.

00:05:06.810 --> 00:05:08.740
So let's actually graph it.

00:05:08.740 --> 00:05:10.910
Let's actually graph
this thing over here.

00:05:10.910 --> 00:05:13.570
So the y's go between
negative 2 and 2.

00:05:13.570 --> 00:05:18.650
The x's go from 1/25th
all the way to 25.

00:05:18.650 --> 00:05:22.680
So let's graph it.

00:05:22.680 --> 00:05:30.050
So that is my y-axis,
and this is my x-axis.

00:05:30.050 --> 00:05:32.116
Draw it like that.

00:05:32.116 --> 00:05:34.190
That is my x-axis.

00:05:34.190 --> 00:05:37.340
And then the y's start at 0.

00:05:37.340 --> 00:05:42.520
Then, you get to
positive 1, positive 2.

00:05:42.520 --> 00:05:44.940
And then you have negative 1.

00:05:44.940 --> 00:05:47.230
And you have negative 2.

00:05:47.230 --> 00:05:49.720
And then on the x-axis,
it's all positive.

00:05:49.720 --> 00:05:53.180
And I'll let you think about
whether the domain here

00:05:53.180 --> 00:05:56.080
is-- well, when you
think about it--

00:05:56.080 --> 00:05:58.270
is a logarithmic
function defined

00:05:58.270 --> 00:06:03.050
for an x that is not positive?

00:06:03.050 --> 00:06:07.190
So is there any power that I can
raise five to that I can get 0?

00:06:07.190 --> 00:06:08.370
No.

00:06:08.370 --> 00:06:11.120
You could raise five to an
infinitely negative power

00:06:11.120 --> 00:06:13.720
to get a very, very, very, very
small number that approaches

00:06:13.720 --> 00:06:15.840
zero, but you can
never get-- there's

00:06:15.840 --> 00:06:18.260
no power that you can
raise 5 to to get 0.

00:06:18.260 --> 00:06:19.799
So x cannot be 0.

00:06:19.799 --> 00:06:21.590
And there's no power
then you could raise 5

00:06:21.590 --> 00:06:24.030
to get another negative number.

00:06:24.030 --> 00:06:25.785
So x can also not be
a negative number.

00:06:25.785 --> 00:06:28.160
So the domain of this function
right over here-- and this

00:06:28.160 --> 00:06:30.410
is relevant, because we want
to think about what we're

00:06:30.410 --> 00:06:33.500
graphing-- the domain here is
x has to be greater than zero.

00:06:33.500 --> 00:06:35.230
Let me write that down.

00:06:35.230 --> 00:06:39.675
The domain here is that x
has to be greater than 0.

00:06:39.675 --> 00:06:41.300
So we're only going
to be able to graph

00:06:41.300 --> 00:06:45.660
this function in
the positive x-axis.

00:06:45.660 --> 00:06:48.380
So with that out of the
way, x gets as large as 25.

00:06:48.380 --> 00:06:51.150
So let me graph-- we
put those points here.

00:06:51.150 --> 00:06:57.430
So that is 5, 10,
15, 20, and 25.

00:06:57.430 --> 00:06:58.884
And then let's plot these.

00:06:58.884 --> 00:07:00.050
So the first one is in blue.

00:07:00.050 --> 00:07:02.720
When x is 1/25 and
y is negative 2--

00:07:02.720 --> 00:07:06.090
When x is 1/25 so
1 is there-- 1/25

00:07:06.090 --> 00:07:09.940
is going to be really close to
there-- Then y is negative 2.

00:07:09.940 --> 00:07:12.680
So it's going to be
like right over there,

00:07:12.680 --> 00:07:14.190
not quite at the y-axis.

00:07:14.190 --> 00:07:17.040
We're at 1/25 to the
right of the y-axis.

00:07:17.040 --> 00:07:17.996
But pretty close.

00:07:17.996 --> 00:07:19.120
So that's right over there.

00:07:19.120 --> 00:07:23.680
That is 1 over 25, comma
negative 2 right over there.

00:07:23.680 --> 00:07:26.270
Then, when x is one
fifth, which is slightly

00:07:26.270 --> 00:07:30.550
further to the right, one
fifth y is negative 1.

00:07:30.550 --> 00:07:32.880
So right over there.

00:07:32.880 --> 00:07:36.950
So this is one
fifth, negative 1.

00:07:36.950 --> 00:07:40.290
Then when x is 1, y is 0.

00:07:40.290 --> 00:07:43.310
So 1 might be right there.

00:07:43.310 --> 00:07:46.550
So this is the point 1,0.

00:07:46.550 --> 00:07:50.630
And then when x is 5, y is 1.

00:07:50.630 --> 00:07:53.180
When x is 5, I covered it
over here, when this is five,

00:07:53.180 --> 00:07:56.530
y is 1.

00:07:56.530 --> 00:07:59.280
So that's the point 5,1.

00:07:59.280 --> 00:08:02.200
And then finally,
when x is 25, y is 2.

00:08:08.060 --> 00:08:11.140
So this is 25,2.

00:08:11.140 --> 00:08:13.050
And then I can
graph the function.

00:08:13.050 --> 00:08:17.030
And I'll do it-- let me do it
in a color-- I'll use this pink.

00:08:17.030 --> 00:08:23.550
So as x gets super, super,
super, super small, y goes

00:08:23.550 --> 00:08:25.820
to negative infinity.

00:08:25.820 --> 00:08:30.490
It gets really small-- to
get x's or as x becomes--

00:08:30.490 --> 00:08:34.179
if you say what power do
you have to raise 5 to

00:08:34.179 --> 00:08:36.539
to get 0.0001?

00:08:36.539 --> 00:08:38.530
It has to be very, very,
very negative power.

00:08:38.530 --> 00:08:43.090
So y is going to be very
negative as we approach 0.

00:08:43.090 --> 00:08:47.390
And then it kind of
moves up like that.

00:08:47.390 --> 00:08:53.110
And then starts to kind of
curve to the right like that.

00:08:53.110 --> 00:08:54.840
And this thing
right over here, is

00:08:54.840 --> 00:08:58.890
going to keep going down at
a steeper and steeper rate.

00:08:58.890 --> 00:09:03.160
And it's never going
to quite touch.

00:09:03.160 --> 00:09:04.060
the y-axis.

00:09:04.060 --> 00:09:06.370
It's going to get closer
and closer to the y-axis.

00:09:06.370 --> 00:09:09.840
But it's never going
to be quite touch it.


Solving logarithmic equations

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:07.800
Solve 4 is equal to
log base b of 81 for b.

00:00:07.800 --> 00:00:11.070
So let's just remind ourselves
what this equation is saying.

00:00:11.070 --> 00:00:15.750
This is saying, if I raise
b to the fourth power,

00:00:15.750 --> 00:00:17.140
then I'm going to get 81.

00:00:17.140 --> 00:00:18.150
Let me rewrite that.

00:00:18.150 --> 00:00:22.890
So if I raise b, that's our
base, so if I raise our base,

00:00:22.890 --> 00:00:25.590
b-- I guess that's why
they picked b. b for base.

00:00:25.590 --> 00:00:29.030
If I raise b to
the fourth power--

00:00:29.030 --> 00:00:31.530
I'll do the fourth
power in orange.

00:00:31.530 --> 00:00:34.010
If I raise b to
the fourth power,

00:00:34.010 --> 00:00:36.690
that is going to be equal to 81.

00:00:36.690 --> 00:00:38.800
I've just rewritten
this equation,

00:00:38.800 --> 00:00:41.250
this logarithmic equation,
I've rewritten it

00:00:41.250 --> 00:00:43.410
as an exponential equation.

00:00:43.410 --> 00:00:45.770
So it says b to the fourth
power is equal to 81,

00:00:45.770 --> 00:00:49.820
and we need to
still solve for b.

00:00:49.820 --> 00:00:51.780
So you just have to
think, what number do I

00:00:51.780 --> 00:00:57.000
have to multiply by itself
four times in order to get 81.

00:00:57.000 --> 00:01:00.080
And 81 might jump out of
you, it is a perfect square.

00:01:00.080 --> 00:01:04.384
We know that 9 times
9 is equal to 81.

00:01:04.384 --> 00:01:05.800
Or we know that
another way to say

00:01:05.800 --> 00:01:08.210
that is, nine squared
is equal to 81.

00:01:08.210 --> 00:01:10.510
But we have to raise
something to the fourth power.

00:01:10.510 --> 00:01:12.980
But 9 itself is 3 times 3.

00:01:15.720 --> 00:01:19.480
So one 9 is 3 times 3, and then
you multiply it by another 9.

00:01:19.480 --> 00:01:22.090
That's another 3 times 3.

00:01:22.090 --> 00:01:24.910
That will also be equal to 81.

00:01:24.910 --> 00:01:25.970
And we can verify it.

00:01:25.970 --> 00:01:31.950
3 times 3 is 9, 9 times 3
is 27, 27 times 3 is 81.

00:01:31.950 --> 00:01:33.680
So this is 3 to
the fourth power.

00:01:33.680 --> 00:01:39.582
9 squared is the same thing
as 3 to the fourth power.

00:01:39.582 --> 00:01:40.540
So there we've done it.

00:01:40.540 --> 00:01:43.070
We've said, well, some
number to the fourth power

00:01:43.070 --> 00:01:43.900
is equal to 81.

00:01:43.900 --> 00:01:47.300
We know that 3 to the
fourth power is equal to 81.

00:01:47.300 --> 00:01:54.270
So we know that b is equal to 3.

00:01:54.270 --> 00:01:55.640
3 to the fourth power is 81.

00:01:55.640 --> 00:01:59.800
Or we could say, log
base three of 81,

00:01:59.800 --> 00:02:03.307
this is saying what power do I
have to raise 3 to, to get 81?

00:02:03.307 --> 00:02:03.890
Well, we know.

00:02:03.890 --> 00:02:08.860
You have to raise 3 to the
fourth power to get to 81.

00:02:08.860 --> 00:02:10.669
And if you know about
fractional exponents,

00:02:10.669 --> 00:02:13.210
and don't worry about this if
this confuses you a little bit,

00:02:13.210 --> 00:02:14.990
you could raise both
of these, if you

00:02:14.990 --> 00:02:17.720
know about fractional exponents
and exponent properties,

00:02:17.720 --> 00:02:19.700
you could do it
this way as well.

00:02:19.700 --> 00:02:22.200
You could take, so you
have b to the fourth power

00:02:22.200 --> 00:02:24.250
is equal to 81.

00:02:24.250 --> 00:02:27.224
You could raise both of
these to the 1/4 power.

00:02:27.224 --> 00:02:28.890
Anything you do to
one side of equation,

00:02:28.890 --> 00:02:30.346
you have to do to
the other side.

00:02:30.346 --> 00:02:31.720
And from our
exponent properties,

00:02:31.720 --> 00:02:33.720
you know that if you raise
something to a power,

00:02:33.720 --> 00:02:35.340
then raise that
to a power, that's

00:02:35.340 --> 00:02:38.160
like raising it to
the 4 times 1/4 power.

00:02:38.160 --> 00:02:41.000
Or this is, essentially, just
raising it to the first power.

00:02:41.000 --> 00:02:43.710
So on the left hand side,
you're just left with b,

00:02:43.710 --> 00:02:46.690
and on the right hand side,
you're left-- is equal to 81

00:02:46.690 --> 00:02:48.310
to the 1/4 power.

00:02:48.310 --> 00:02:50.490
But figuring out what
81 to the 1/4 power,

00:02:50.490 --> 00:02:52.680
you really have to go
through this exercise anyway.

00:02:52.680 --> 00:02:55.030
Because when you raise
something to the 1/4 power,

00:02:55.030 --> 00:02:58.275
you're really saying, well,
what do I have to multiply?

00:02:58.275 --> 00:03:00.650
What do I have to raise to
the fourth power to get to 81?

00:03:00.650 --> 00:03:03.156
And then you get what
81 is to the 1/4 power.

00:03:03.156 --> 00:03:05.030
So actually, this is
another way of realizing

00:03:05.030 --> 00:03:08.530
that 81 to the 1/4
power is equal to 3.

00:03:08.530 --> 00:03:10.600
3 to the fourth
power is equal to 81.

00:03:10.600 --> 00:03:13.210
81 to the 1/4 power
is equal to 3.

00:03:13.210 --> 00:03:17.670
But if this is confusing to you,
don't worry about it for now.

00:03:17.670 --> 00:03:19.400
The important thing
is you understand

00:03:19.400 --> 00:03:22.720
what the logarithm
is actually saying.

00:03:22.720 --> 00:03:26.660
If I raise b to the
fourth power I get 81.


Logarithmic Equations 2

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.985
We're asked to convert 10 to the second power,

00:00:02.985 --> 00:00:08.104
or 10 squared, equals 100 into a logarithmic equation.

00:00:08.104 --> 00:00:12.817
Now this is saying, if I take base 10-- so I'll do 10 in yellow --

00:00:12.817 --> 00:00:19.770
I take the base 10 to the exponent 2, then I get 100.

00:00:19.770 --> 00:00:23.567
When you do a logarithm, it is really kind of the inverse of that,

00:00:23.567 --> 00:00:25.840
it is really kind of the other way of thinking about it.

00:00:25.840 --> 00:00:37.361
A logarithm says: "what power do I have to raise the base to, to get 100?"

00:00:39.023 --> 00:00:45.372
So if I were to write log, base 10, of 100-- and I wrote the ten a little bit bigger than is typically done,

00:00:45.372 --> 00:00:50.037
let me write it this way-- log base 10 of 100, this says

00:00:50.037 --> 00:00:55.769
what power do I have to raise 10 to, my base, to get 100?

00:00:55.769 --> 00:00:58.914
And we know from this, and probably you knew this even if we didn't have this written down,

00:00:58.914 --> 00:01:01.839
you would have to raise it to the second power.

00:01:01.839 --> 00:01:06.316
So this is equal to 2.

00:01:06.316 --> 00:01:11.838
So right over here we are saying in an exponential equation, the base ten, to the second power

00:01:11.838 --> 00:01:15.505
is equal to 100. Here in the logarithmic equation we are

00:01:15.505 --> 00:01:22.105
saying that the power that we have to raise 10 to to get 100 is equal to 2.

00:01:22.105 --> 00:01:27.371
So these are really saying the exact same thing.

00:01:27.371 --> 00:01:31.304
And just so you get the terminology right, this, in both cases,

00:01:31.304 --> 00:01:35.706
the ten is referred to as the base, this is the base,

00:01:35.706 --> 00:01:39.234
and in both cases the 2, -- let me write it--

00:01:39.234 --> 00:01:47.402
so in both cases the 2 is referred to, and I think you know this already, as the exponent.

00:01:47.402 --> 00:01:52.105
And then the 100 is what you get when you raise the base to the exponent.

00:01:52.105 --> 00:01:56.105
So these are two ways of writing the exact same truth about 10 to the second power equaling 100.


Logarithmic equations

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:05.450
Convert log base
4 of 2 equals 1/2

00:00:05.450 --> 00:00:09.050
into an exponential equation.

00:00:09.050 --> 00:00:10.990
So a logarithm
literally tells you--

00:00:10.990 --> 00:00:13.480
so if we were to just look
at this part right over here,

00:00:13.480 --> 00:00:16.010
a logarithm is just saying,
tell me the exponent

00:00:16.010 --> 00:00:18.620
that I would have
to raise my base to,

00:00:18.620 --> 00:00:22.040
that I would have to
raise 4 to, to get to 2.

00:00:22.040 --> 00:00:23.990
So tell me the
exponent that I would

00:00:23.990 --> 00:00:26.190
have to raise it to to get to 2.

00:00:26.190 --> 00:00:29.460
And so this equation written
as a logarithmic equation says,

00:00:29.460 --> 00:00:37.171
I need to raise 4 to the
1/2 power in order to get 2.

00:00:37.171 --> 00:00:39.420
So if I wanted to write it
as an exponential equation,

00:00:39.420 --> 00:00:42.530
I could literally just
say, well, if I raise 4--

00:00:42.530 --> 00:00:44.150
and that's this 4
right over here.

00:00:44.150 --> 00:00:48.740
That's my base-- to this
exponent, to the 1/2 power,

00:00:48.740 --> 00:00:53.830
then that is going
to be equal to 2.

00:00:53.830 --> 00:00:54.520
And we're done.

00:00:54.520 --> 00:00:58.110
We've written the same equation
as an exponential equation.

00:00:58.110 --> 00:01:00.850
The value of a logarithm--
this is saying,

00:01:00.850 --> 00:01:03.320
what do I get if I raise
4 to the 1/2 power?

00:01:03.320 --> 00:01:06.390
I get 2 if I raise
4 to the 1/2 power.

00:01:06.390 --> 00:01:12.210
This is saying, what power do I
have to raise 4 to to get to 2?

00:01:12.210 --> 00:01:14.800
And then that is equal to 1/2.

00:01:14.800 --> 00:01:16.960
So a logarithm is
saying, what power

00:01:16.960 --> 00:01:18.530
do I have to raise
the base to to get

00:01:18.530 --> 00:01:20.210
what's inside the logarithm?

00:01:20.210 --> 00:01:25.738
This says, what do I get to
when I raise 4 to the 1/2 power?


Graphing exponential functions

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:05.910
We're asked to graph y is
equal to 5 to the x-th power.

00:00:05.910 --> 00:00:08.060
And we'll just do this
the most basic way.

00:00:08.060 --> 00:00:10.100
We'll just try out
some values for x

00:00:10.100 --> 00:00:11.500
and see what we get for y.

00:00:11.500 --> 00:00:13.950
And then we'll plot
those coordinates.

00:00:13.950 --> 00:00:17.180
So let's try some negative
and some positive values.

00:00:17.180 --> 00:00:18.840
And I'll try to
center them around 0.

00:00:18.840 --> 00:00:20.840
So this will be my x values.

00:00:20.840 --> 00:00:22.760
This will be my y values.

00:00:22.760 --> 00:00:25.480
Let's start first with something
reasonably negative but not

00:00:25.480 --> 00:00:26.140
too negative.

00:00:26.140 --> 00:00:29.920
So let's say we start with
x is equal to negative 2.

00:00:29.920 --> 00:00:32.400
Then y is equal to
5 to the x power,

00:00:32.400 --> 00:00:36.280
or 5 to the negative
2 power, which we know

00:00:36.280 --> 00:00:41.290
is the same thing as 1 over 5
to the positive 2 power, which

00:00:41.290 --> 00:00:44.174
is just 1/25.

00:00:44.174 --> 00:00:45.340
Now let's try another value.

00:00:45.340 --> 00:00:48.000
What happens when x is
equal to negative 1?

00:00:48.000 --> 00:00:51.920
Then y is 5 to the
negative 1 power, which

00:00:51.920 --> 00:00:58.460
is the same thing as 1 over 5
to the first power, or just 1/5.

00:00:58.460 --> 00:01:00.780
Now let's think about
when x is equal to 0.

00:01:00.780 --> 00:01:04.769
Then y is going to be equal
to 5 to the 0-th power, which

00:01:04.769 --> 00:01:06.580
we know anything
to the 0-th power

00:01:06.580 --> 00:01:08.910
is going to be equal to 1.

00:01:08.910 --> 00:01:12.480
So this is going
to be equal to 1.

00:01:12.480 --> 00:01:14.670
And then finally,
we have-- well,

00:01:14.670 --> 00:01:18.560
actually, let's try a
couple of more points here.

00:01:18.560 --> 00:01:21.240
Let me extend this table
a little bit further.

00:01:21.240 --> 00:01:24.140
Let's try out x is equal to 1.

00:01:24.140 --> 00:01:28.520
Then y is 5 to the first power,
which is just equal to 5.

00:01:28.520 --> 00:01:31.340
And let's do one
last value over here.

00:01:31.340 --> 00:01:33.190
Let's see what happens
when x is equal to 2.

00:01:33.190 --> 00:01:37.350
Then y is 5 squared,
5 to the second power,

00:01:37.350 --> 00:01:39.650
which is just equal to 25.

00:01:39.650 --> 00:01:44.660
And now we can plot it to
see how this actually looks.

00:01:44.660 --> 00:01:47.670
So let me get some
graph paper going here.

00:01:47.670 --> 00:01:51.410
My x's go as low as negative
2, as high as positive 2.

00:01:51.410 --> 00:01:57.590
And then my y's go all the way
from 1/25 all the way to 25.

00:01:57.590 --> 00:02:00.560
So I have positive
values over here.

00:02:00.560 --> 00:02:02.490
So let me draw it like this.

00:02:02.490 --> 00:02:06.450
So this could be my x-axis.

00:02:06.450 --> 00:02:08.610
That could be my x-axis.

00:02:08.610 --> 00:02:11.715
And then let's make
this my y-axis.

00:02:11.715 --> 00:02:14.710
I'll draw it as neatly as I can.

00:02:14.710 --> 00:02:16.940
So let's make that my y-axis.

00:02:16.940 --> 00:02:20.150
And my x values, this
could be negative 2.

00:02:20.150 --> 00:02:22.570
Actually, make my
y-axis keep going.

00:02:22.570 --> 00:02:24.000
So that's y.

00:02:24.000 --> 00:02:25.170
This is x.

00:02:25.170 --> 00:02:26.330
That's a negative 2.

00:02:26.330 --> 00:02:28.300
That's negative 1.

00:02:28.300 --> 00:02:29.670
That's 0.

00:02:29.670 --> 00:02:31.310
That is 1.

00:02:31.310 --> 00:02:33.690
And that is positive 2.

00:02:33.690 --> 00:02:35.350
And let's plot the points.

00:02:35.350 --> 00:02:37.970
x is negative 2. y is 1/25.

00:02:37.970 --> 00:02:41.040
Actually, let me make
the scale on the y-axis.

00:02:41.040 --> 00:02:42.396
So let's make this.

00:02:42.396 --> 00:02:44.020
So we're going to go
all the way to 25.

00:02:44.020 --> 00:02:46.235
So let's say that this is 5.

00:02:46.235 --> 00:02:48.610
Actually, I have to do it a
little bit smaller than that,

00:02:48.610 --> 00:02:49.250
too.

00:02:49.250 --> 00:02:55.650
So this is going to
be 5, 10, 15, 20.

00:02:55.650 --> 00:02:59.530
And then 25 would be right where
I wrote the y, give or take.

00:02:59.530 --> 00:03:00.780
So now let's plot them.

00:03:00.780 --> 00:03:01.773
Negative 2, 1/25.

00:03:04.470 --> 00:03:05.700
1 is going to be like there.

00:03:05.700 --> 00:03:08.670
So 1/25 is going to be really,
really close to the x-axis.

00:03:08.670 --> 00:03:09.830
That's about 1/25.

00:03:09.830 --> 00:03:13.772
So that is negative 2, 1/25.

00:03:13.772 --> 00:03:15.230
It's not going to
be on the x-axis.

00:03:15.230 --> 00:03:16.710
1/25 is obviously
greater than 0.

00:03:16.710 --> 00:03:19.240
It's going to be really,
really, really, really, close.

00:03:19.240 --> 00:03:23.040
Now let's do this point here
in orange, negative 1, 1/5.

00:03:23.040 --> 00:03:27.750
Negative 1/5-- 1/5 on this
scale is still pretty close.

00:03:27.750 --> 00:03:29.700
It's pretty close.

00:03:29.700 --> 00:03:33.620
So that right over there
is negative 1, 1/5.

00:03:33.620 --> 00:03:36.680
And now in blue,
we have 0 comma 1.

00:03:36.680 --> 00:03:40.150
0 comma 1 is going to
be right about there.

00:03:40.150 --> 00:03:43.280
If this is 2 and 1/2, that
looks about right for 1.

00:03:43.280 --> 00:03:45.590
And then we have 1 comma 5.

00:03:45.590 --> 00:03:49.660
1 comma 5 puts us
right over there.

00:03:49.660 --> 00:03:51.710
And then finally,
we have 2 comma 25.

00:03:51.710 --> 00:03:54.730
When x is 2, y is 25.

00:03:54.730 --> 00:04:00.420
2 comma 25 puts us
right about there.

00:04:00.420 --> 00:04:03.140
And so I think you see what
happens with this function,

00:04:03.140 --> 00:04:05.640
with this graph.

00:04:05.640 --> 00:04:10.010
The further in the
negative direction we go,

00:04:10.010 --> 00:04:12.720
5 to ever-increasing
negative powers

00:04:12.720 --> 00:04:15.440
gets closer and closer
to 0, but never quite.

00:04:15.440 --> 00:04:18.706
So we're leaving 0, getting
slightly further, further,

00:04:18.706 --> 00:04:20.230
further from 0.

00:04:20.230 --> 00:04:25.235
Right at the y-axis,
we have y equal 1.

00:04:28.320 --> 00:04:31.600
Right at x is equal to 0,
we have y is equal to 1.

00:04:31.600 --> 00:04:34.960
And then once x starts
increasing beyond 0,

00:04:34.960 --> 00:04:37.310
then we start seeing what
the exponential is good at,

00:04:37.310 --> 00:04:39.197
which is just this
very rapid increase.

00:04:39.197 --> 00:04:41.280
Some people would call it
an exponential increase,

00:04:41.280 --> 00:04:43.400
which is obviously the
case right over here.

00:04:43.400 --> 00:04:46.140
So then if I just
keep this curve going,

00:04:46.140 --> 00:04:48.970
you see it's just going
on this sometimes called

00:04:48.970 --> 00:04:50.140
a hockey stick.

00:04:50.140 --> 00:04:51.980
It just keeps on
going up like this

00:04:51.980 --> 00:04:55.000
at a super fast rate,
ever-increasing rate.

00:04:55.000 --> 00:04:57.000
So you could keep going
forever to the left,

00:04:57.000 --> 00:04:58.958
and you'd get closer and
closer and closer to 0

00:04:58.958 --> 00:05:00.550
without quite getting to 0.

00:05:00.550 --> 00:05:03.202
So 5 to the negative
billionth power

00:05:03.202 --> 00:05:04.660
is still not going
to get you to 0,

00:05:04.660 --> 00:05:06.890
but it's going to get you
pretty darn close to 0.

00:05:06.890 --> 00:05:09.490
But obviously, if you go to 5
to the positive billionth power,

00:05:09.490 --> 00:05:11.260
you're going to get
to a super huge number

00:05:11.260 --> 00:05:13.990
because this thing is just
going to keep skyrocketing up

00:05:13.990 --> 00:05:14.491
like that.

00:05:14.491 --> 00:05:16.198
So let me just draw
the whole curve, just

00:05:16.198 --> 00:05:17.210
to make sure you see it.

00:05:17.210 --> 00:05:19.940
Over here, I'm not actually on
0, although the way I drew it,

00:05:19.940 --> 00:05:20.940
it might look like that.

00:05:20.940 --> 00:05:21.910
I'm slightly above 0.

00:05:21.910 --> 00:05:24.770
I'm increasing above that,
increasing above that.

00:05:24.770 --> 00:05:27.210
And once I get into the
positive x's, then I

00:05:27.210 --> 00:05:31.270
start really,
really shooting up.


Quotient of functions

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:05.330
f of x is equal to 2x
squared plus 15x minus 8.

00:00:05.330 --> 00:00:09.690
g of x is equal to x
squared plus 10x plus 16.

00:00:09.690 --> 00:00:12.510
Find f/g of x.

00:00:12.510 --> 00:00:16.030
Or you could interpret this
is as f divided by g of x.

00:00:16.030 --> 00:00:18.110
And so based on the
way I just said it,

00:00:18.110 --> 00:00:20.670
you have a sense
of what this means.

00:00:20.670 --> 00:00:26.230
f/g, or f divided by g, of x,
by definition, this is just

00:00:26.230 --> 00:00:31.230
another way to write f
of x divided by g of x.

00:00:31.230 --> 00:00:33.440
You could view this as
a function, a function

00:00:33.440 --> 00:00:36.580
of x that's defined
by dividing f

00:00:36.580 --> 00:00:39.290
of x by g of x, by creating
a rational expression where

00:00:39.290 --> 00:00:42.850
f of x is in the numerator and
g of x is in the denominator.

00:00:42.850 --> 00:00:47.490
And so this is going to be equal
to f of x-- we have right up

00:00:47.490 --> 00:00:54.020
here-- is 2x
squared 15x minus 8.

00:00:54.020 --> 00:00:59.050
And g of x-- I will do in blue--
is right over here, g of x.

00:00:59.050 --> 00:01:01.260
So this is all
going to be over g

00:01:01.260 --> 00:01:08.880
of x, which is x squared
plus 10x plus 16.

00:01:08.880 --> 00:01:11.130
And you could leave it this
way, or you could actually

00:01:11.130 --> 00:01:14.800
try to simplify
this a little bit.

00:01:14.800 --> 00:01:16.550
And the easiest way
to simplify this

00:01:16.550 --> 00:01:19.600
would see if we could factor the
numerator and the denominator

00:01:19.600 --> 00:01:22.110
expressions into maybe
simpler expressions.

00:01:22.110 --> 00:01:25.040
And maybe some of them might be
on-- maybe both the numerator

00:01:25.040 --> 00:01:27.810
and denominator is divisible
by the same expression.

00:01:27.810 --> 00:01:29.500
So let's try to
factor each of them.

00:01:29.500 --> 00:01:31.030
So first, let's
try the numerator.

00:01:31.030 --> 00:01:32.660
And I'll actually do it up here.

00:01:32.660 --> 00:01:33.320
So let's do it.

00:01:33.320 --> 00:01:34.611
Actually, I'll do it down here.

00:01:34.611 --> 00:01:39.110
So if I'm looking at 2x
squared plus 15x minus 8,

00:01:39.110 --> 00:01:44.220
we have a quadratic expression
where the coefficient is not 1.

00:01:44.220 --> 00:01:47.230
And so one technique to factor
this is to factor by grouping.

00:01:47.230 --> 00:01:49.310
You could also use
the quadratic formula.

00:01:49.310 --> 00:01:51.140
And when you factor
by grouping, you're

00:01:51.140 --> 00:01:54.170
going to split up
this term, this 15x.

00:01:54.170 --> 00:01:57.830
And you're going to
split up into two terms

00:01:57.830 --> 00:01:59.850
where the coefficients
are, if I were

00:01:59.850 --> 00:02:03.090
to take the product
of those coefficients,

00:02:03.090 --> 00:02:04.800
they're going to be
equal to the product

00:02:04.800 --> 00:02:06.700
of the first and the last terms.

00:02:06.700 --> 00:02:09.432
And we proved that
in other videos.

00:02:09.432 --> 00:02:11.390
So essentially, we want
to think of two numbers

00:02:11.390 --> 00:02:14.750
that add up to 15,
but whose product

00:02:14.750 --> 00:02:17.600
is equal to negative 16.

00:02:17.600 --> 00:02:19.940
And this is just the technique
of factoring by grouping.

00:02:19.940 --> 00:02:23.430
It's really just an attempt to
simplify this right over here.

00:02:23.430 --> 00:02:26.650
So what two numbers that,
if I take their product,

00:02:26.650 --> 00:02:28.270
I get negative 16.

00:02:28.270 --> 00:02:30.091
But if I add them, I get 15?

00:02:30.091 --> 00:02:32.340
Well, if I take the product
and get a negative number,

00:02:32.340 --> 00:02:34.290
that means they have to
have a different sign.

00:02:34.290 --> 00:02:36.540
And so that means one of
them is going to be positive,

00:02:36.540 --> 00:02:37.730
one of them is going
to be negative,

00:02:37.730 --> 00:02:38.890
which means one of them
is going to be larger

00:02:38.890 --> 00:02:41.500
than 15 and one of them is
going to be smaller than 15.

00:02:41.500 --> 00:02:43.170
And the most obvious
one there might

00:02:43.170 --> 00:02:46.960
be 16, positive
16, and negative 1.

00:02:46.960 --> 00:02:49.900
If I multiply these two things,
I definitely get negative 16.

00:02:49.900 --> 00:02:55.040
If I add these two things,
I definitely get 15.

00:02:55.040 --> 00:02:57.240
So what we can do is
we can split this.

00:02:57.240 --> 00:03:02.450
We can rewrite this expression
as 2x squared plus 2x squared

00:03:02.450 --> 00:03:10.400
plus 16x minus x minus 8.

00:03:10.400 --> 00:03:14.910
All I did here is I
took this middle term

00:03:14.910 --> 00:03:16.800
and, using this technique
right over here,

00:03:16.800 --> 00:03:19.920
I split it into
16x minus x, which

00:03:19.920 --> 00:03:22.140
is clearly still just 15x.

00:03:22.140 --> 00:03:24.010
Now what's useful
about this-- and this

00:03:24.010 --> 00:03:27.280
is why we call it factoring
by grouping-- is we can see,

00:03:27.280 --> 00:03:29.840
are there any common factors
in these first two terms

00:03:29.840 --> 00:03:31.110
right over here?

00:03:31.110 --> 00:03:35.050
Well, both 2x squared and 16x,
they are both divisible by 2x.

00:03:35.050 --> 00:03:38.400
So you could factor out a
2x of these first two terms.

00:03:38.400 --> 00:03:45.622
This is the same thing as
2x times x plus x plus 8.

00:03:45.622 --> 00:03:49.610
16 divided by 2 is 8,
x divided by x is 1.

00:03:49.610 --> 00:03:52.330
So this is 2x times x plus 8.

00:03:52.330 --> 00:03:54.940
And then the second two
terms right over here--

00:03:54.940 --> 00:03:56.440
this is the whole
basis of factoring

00:03:56.440 --> 00:03:59.450
by grouping-- we can
factor out a negative 1.

00:03:59.450 --> 00:04:02.990
So this is equal to
negative 1 times x plus 8.

00:04:02.990 --> 00:04:06.070
And what's neat here is
now we have two terms.

00:04:06.070 --> 00:04:07.930
Both of them have
an x plus 8 in them.

00:04:07.930 --> 00:04:10.480
So we can factor
out an x plus 8.

00:04:10.480 --> 00:04:15.750
So if we factor out
an x plus 8, we're

00:04:15.750 --> 00:04:23.380
left with 2x minus 1, put
parentheses around it,

00:04:23.380 --> 00:04:26.760
times the factored out x plus 8.

00:04:26.760 --> 00:04:28.470
So we've simplified
the numerator.

00:04:28.470 --> 00:04:29.874
The numerator can be rewritten.

00:04:29.874 --> 00:04:32.290
And you could have gotten here
using the quadratic formula

00:04:32.290 --> 00:04:33.120
as well.

00:04:33.120 --> 00:04:38.120
The numerator is 2x
minus 1 times x plus 8.

00:04:38.120 --> 00:04:40.230
And now see if you can
factor the denominator.

00:04:40.230 --> 00:04:41.730
And this one's more
straightforward.

00:04:41.730 --> 00:04:42.930
The coefficient here is 1.

00:04:42.930 --> 00:04:44.790
So we just have to
think of two numbers

00:04:44.790 --> 00:04:47.200
that when I multiply
them, I get 16.

00:04:47.200 --> 00:04:48.850
And when I add them, I get 10.

00:04:48.850 --> 00:04:53.690
And the obvious one is 8 and
2, positive 8 and positive 2.

00:04:53.690 --> 00:05:01.160
So we can write this as
x plus 2 times x plus 8.

00:05:01.160 --> 00:05:03.670
And now, we can simplify it.

00:05:03.670 --> 00:05:07.220
We can divide the numerator
and denominator by x plus 8,

00:05:07.220 --> 00:05:11.139
assuming that x does
not equal negative 8.

00:05:11.139 --> 00:05:12.680
Because this function
right over here

00:05:12.680 --> 00:05:14.710
that's defined by
f divided by g,

00:05:14.710 --> 00:05:17.620
it is not defined when
g of x is equal to 0,

00:05:17.620 --> 00:05:20.660
because then you have
something divided by 0.

00:05:20.660 --> 00:05:22.540
And the only times that
g of x is equal to 0

00:05:22.540 --> 00:05:25.620
is when x is equal to negative
2 or x is equal to negative 8.

00:05:25.620 --> 00:05:28.390
So if we divide the numerator
and the denominator by x plus 8

00:05:28.390 --> 00:05:32.880
to simplify it, in order to not
change the function definition,

00:05:32.880 --> 00:05:35.660
we have to still put the
constraint that x cannot be

00:05:35.660 --> 00:05:36.989
equal to negative 8.

00:05:36.989 --> 00:05:39.280
That the original function,
in order to not change it--

00:05:39.280 --> 00:05:41.321
because if I just cancelled
these two things out,

00:05:41.321 --> 00:05:43.860
the new function with these
canceled would be defined when

00:05:43.860 --> 00:05:45.276
x is equal to negative 8.

00:05:45.276 --> 00:05:46.650
But we want this
simplified thing

00:05:46.650 --> 00:05:48.570
to be the same exact function.

00:05:48.570 --> 00:05:51.100
And this exact
function is not defined

00:05:51.100 --> 00:05:53.000
when x is equal to negative 8.

00:05:53.000 --> 00:05:57.870
So now we can write f/g
of x, which is really just

00:05:57.870 --> 00:06:01.900
f of x divided by g of
x, is equal to 2x minus 1

00:06:01.900 --> 00:06:03.720
over x plus 2.

00:06:03.720 --> 00:06:07.380
You have to put the condition
there that x cannot be equal

00:06:07.380 --> 00:06:08.470
to negative 8.

00:06:08.470 --> 00:06:10.280
If you lost this
condition, then it

00:06:10.280 --> 00:06:12.230
won't be the exact
same function as this,

00:06:12.230 --> 00:06:17.009
because this is not defined
when x is equal to negative 8.


Product of functions

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.750
f of x is equal to 7x minus 5.

00:00:03.750 --> 00:00:08.340
g of x is equal to x to
the third power plus 4x.

00:00:08.340 --> 00:00:14.550
And then they ask us to find f
times g of x So the first thing

00:00:14.550 --> 00:00:18.860
to realize is that this
notation f times g of x

00:00:18.860 --> 00:00:20.790
is just referring
to a function that

00:00:20.790 --> 00:00:23.820
is a product of f
of x and g of x.

00:00:23.820 --> 00:00:30.872
So by definition, this notation
just means f of x times g of x.

00:00:30.872 --> 00:00:32.330
And then we just
have to substitute

00:00:32.330 --> 00:00:35.020
f of x with this definition,
g of x with this definition,

00:00:35.020 --> 00:00:39.010
and then multiply out these
algebraic expressions.

00:00:39.010 --> 00:00:41.340
f of x is right over there.

00:00:41.340 --> 00:00:44.920
And g of x, is right over there.

00:00:44.920 --> 00:00:46.200
So let's do it.

00:00:46.200 --> 00:00:48.570
So this is going to be
equal to-- switch back

00:00:48.570 --> 00:00:49.776
to the orange color.

00:00:49.776 --> 00:00:51.400
It's going to be
equal to f of x, which

00:00:51.400 --> 00:00:56.790
is 7x minus 5
times g of x, and g

00:00:56.790 --> 00:01:00.790
of x is x to the
third power plus 4x.

00:01:03.730 --> 00:01:07.209
And you could-- we're
multiplying two expressions

00:01:07.209 --> 00:01:08.250
that each have two terms.

00:01:08.250 --> 00:01:10.050
You could use FOIL if you like.

00:01:10.050 --> 00:01:12.570
I don't like using
FOIL because you

00:01:12.570 --> 00:01:14.260
might forget what
it's even about.

00:01:14.260 --> 00:01:18.290
Foil is really just using the
distributive property twice.

00:01:18.290 --> 00:01:20.870
So for example, you
take this expression.

00:01:20.870 --> 00:01:23.600
Whatever you have out here,
if you had a 9 out here,

00:01:23.600 --> 00:01:25.070
or an a, or an x, or anything.

00:01:25.070 --> 00:01:26.570
Now you have 7x minus 5.

00:01:26.570 --> 00:01:28.640
If you're multiplying it
times this expression,

00:01:28.640 --> 00:01:32.610
you would multiply this
times each term over here.

00:01:32.610 --> 00:01:35.790
So when you multiply 7x
minus 5 times x to the third,

00:01:35.790 --> 00:01:37.930
you get-- I'll
write it this way.

00:01:37.930 --> 00:01:40.470
You get x to the third times--
actually, let me write it

00:01:40.470 --> 00:01:42.010
the other way.

00:01:42.010 --> 00:01:46.300
You get 7x minus 5
times x to the third.

00:01:49.190 --> 00:01:56.480
And then you have plus
7x minus 5 times 4x.

00:01:59.870 --> 00:02:02.240
And now we can do the
distributive property again.

00:02:02.240 --> 00:02:03.616
We're not normally
used to seeing

00:02:03.616 --> 00:02:05.614
the things we distribute
on the right hand side.

00:02:05.614 --> 00:02:06.740
It's the same exact idea.

00:02:06.740 --> 00:02:08.669
We could put the x to
the third here as well.

00:02:08.669 --> 00:02:10.169
And when we distribute,
you multiply

00:02:10.169 --> 00:02:14.300
x to the third times 7x
and times negative 5.

00:02:14.300 --> 00:02:19.820
x to the third times 7x
is 7x to the fourth power.

00:02:19.820 --> 00:02:24.012
X to the third times negative
5 is minus 5x to the third.

00:02:24.012 --> 00:02:25.220
And then you do it over here.

00:02:25.220 --> 00:02:29.070
You distribute the
4x over the 7x.

00:02:29.070 --> 00:02:33.750
4x times 7x is plus 28x squared.

00:02:33.750 --> 00:02:39.827
4x times negative
5 is minus 20x.

00:02:39.827 --> 00:02:41.410
And let's see if we
can simplify this.

00:02:41.410 --> 00:02:44.130
We only have one fourth degree
term, one third degree term,

00:02:44.130 --> 00:02:46.982
one second degree term,
and one first degree term.

00:02:46.982 --> 00:02:48.690
Actually, we can't
simplify this anymore.

00:02:48.690 --> 00:02:49.610
And we're done.

00:02:49.610 --> 00:02:52.700
This is the product of those
two function definitions.

00:02:52.700 --> 00:02:54.440
This is f times g of x.

00:02:54.440 --> 00:02:58.530
It is a new function created
by multiplying the other two

00:02:58.530 --> 00:03:00.080
functions.


Sum of functions

WEBVTT
Kind: captions
Language: en

00:00:00.900 --> 00:00:03.200
f of x is equal to 9 minus x.

00:00:03.200 --> 00:00:05.620
Actually, it should
be 9 minus x squared.

00:00:05.620 --> 00:00:11.080
And g of x is equal to 5x
squared plus 2x plus 1.

00:00:11.080 --> 00:00:15.090
And then they say
find f plus g of x.

00:00:15.090 --> 00:00:16.590
And this looks a
little bit bizarre.

00:00:16.590 --> 00:00:18.190
What kind of notation is this?

00:00:18.190 --> 00:00:19.980
And that's really the
core of this problem

00:00:19.980 --> 00:00:21.830
is just to realize that
when someone writes

00:00:21.830 --> 00:00:26.300
f plus g in parentheses like
this of x this is just notation

00:00:26.300 --> 00:00:29.120
for-- so f plus g of
x, I just rewrote it.

00:00:29.120 --> 00:00:30.430
This is the same thing.

00:00:30.430 --> 00:00:32.830
This is just a kind
of shorthand notation

00:00:32.830 --> 00:00:36.760
for f of x plus g of x.

00:00:36.760 --> 00:00:39.890
You could view f plus g
as a new function that's

00:00:39.890 --> 00:00:42.744
created by adding the
other two functions.

00:00:42.744 --> 00:00:44.410
But when you view it
like this-- so this

00:00:44.410 --> 00:00:45.890
is really what we have to find.

00:00:45.890 --> 00:00:48.190
Then, you just have to
add these two functions.

00:00:48.190 --> 00:00:52.320
So f of x, they've given the
definition right over there,

00:00:52.320 --> 00:00:53.675
is 9 minus x squared.

00:00:56.620 --> 00:01:01.540
And g of x, they've given the
definition right over here,

00:01:01.540 --> 00:01:07.450
is 5x squared plus 2x plus 1.

00:01:07.450 --> 00:01:09.369
So when you add
f of x to g of x,

00:01:09.369 --> 00:01:11.160
this is going to be
equal to-- and I'm just

00:01:11.160 --> 00:01:13.159
rewriting a lot of things
just to make it clear.

00:01:13.159 --> 00:01:17.670
The f of x part is
9 minus x squared.

00:01:17.670 --> 00:01:19.740
And then you have the plus.

00:01:19.740 --> 00:01:21.070
I'll do that in yellow.

00:01:21.070 --> 00:01:28.410
Plus the g of x part, which
is 5x squared plus 2x plus 1.

00:01:28.410 --> 00:01:31.010
And now we can just
simplify this a little bit.

00:01:31.010 --> 00:01:33.526
We can say-- let me just
get rid of the parentheses.

00:01:33.526 --> 00:01:34.400
I'll just rewrite it.

00:01:34.400 --> 00:01:39.050
So this is equal to
9 minus x squared

00:01:39.050 --> 00:01:41.961
plus-- since this
is a positive, we

00:01:41.961 --> 00:01:43.710
don't have to worry
about the parentheses.

00:01:43.710 --> 00:01:48.910
So plus 5x squared
plus 2x plus 1.

00:01:48.910 --> 00:01:52.020
And then we have two x squared
terms or second degree terms.

00:01:52.020 --> 00:01:53.620
We have 5x squared.

00:01:53.620 --> 00:01:55.300
And then we have
negative x squared.

00:01:55.300 --> 00:01:59.470
5x squared minus x
squared is 4x squared.

00:01:59.470 --> 00:02:02.090
So you get 4x squared when
you combine these two terms.

00:02:02.090 --> 00:02:07.630
You only have one first degree
term 2x there, so plus 2x.

00:02:07.630 --> 00:02:12.220
And then you have two constant
terms, or 0 degree terms.

00:02:12.220 --> 00:02:14.210
So you have the 9
and you have the 1.

00:02:14.210 --> 00:02:17.350
And 9 plus 1 gets us to 10.

00:02:17.350 --> 00:02:19.512
So we're done.

00:02:19.512 --> 00:02:25.600
f plus g of x is equal to
4x squared plus 2x plus 10.

00:02:25.600 --> 00:02:27.430
Notice this is a
new function that's

00:02:27.430 --> 00:02:32.700
created by summing the function
definitions of f and g.


Domain of a radical function

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:03.200
Find the domain
of f of x is equal

00:00:03.200 --> 00:00:06.920
to the principal square
root of 2x minus 8.

00:00:06.920 --> 00:00:08.620
So the domain of
a function is just

00:00:08.620 --> 00:00:12.000
the set of all of the possible
valid inputs into the function,

00:00:12.000 --> 00:00:14.380
or all of the possible
values for which

00:00:14.380 --> 00:00:16.700
the function is defined.

00:00:16.700 --> 00:00:19.340
And when we look at how the
function is defined, right

00:00:19.340 --> 00:00:22.070
over here, as the square root,
the principal square root of 2x

00:00:22.070 --> 00:00:24.790
minus 8, it's only
going to be defined

00:00:24.790 --> 00:00:26.620
when it's taking the
principal square root

00:00:26.620 --> 00:00:28.770
of a non-negative number.

00:00:28.770 --> 00:00:32.940
And so 2x minus
8, it's only going

00:00:32.940 --> 00:00:37.030
to be defined when 2x minus 8
is greater than or equal to 0.

00:00:37.030 --> 00:00:39.530
It can be 0, because then you
just take the square root of 0

00:00:39.530 --> 00:00:40.420
is 0.

00:00:40.420 --> 00:00:41.682
It can be positive.

00:00:41.682 --> 00:00:43.640
But if this was negative,
then all of a sudden,

00:00:43.640 --> 00:00:46.130
this principle square root
function, which we're assuming

00:00:46.130 --> 00:00:48.950
is just the plain vanilla
one for real numbers,

00:00:48.950 --> 00:00:50.510
it would not be defined.

00:00:50.510 --> 00:00:54.260
So this function definition is
only defined when 2x minus 8

00:00:54.260 --> 00:00:57.250
is greater than or equal to 0.

00:00:57.250 --> 00:00:59.500
And then we could
say if 2x minus 8

00:00:59.500 --> 00:01:01.220
has to be greater
than or equal to 0,

00:01:01.220 --> 00:01:02.990
we can solve this
inequality to see

00:01:02.990 --> 00:01:05.630
what it's saying about
what x has to be.

00:01:05.630 --> 00:01:08.650
So if we add 8 to both
sides of this inequality,

00:01:08.650 --> 00:01:12.620
you get-- so let me just
add 8 to both sides.

00:01:12.620 --> 00:01:13.710
These 8's cancel out.

00:01:13.710 --> 00:01:17.451
You get 2x is greater
than or equal to 8.

00:01:17.451 --> 00:01:19.000
0 plus 8 is 8.

00:01:19.000 --> 00:01:21.010
And then you divide
both sides by 2.

00:01:21.010 --> 00:01:22.710
Since 2 is a
positive number, you

00:01:22.710 --> 00:01:24.610
don't have to swap
the inequality.

00:01:24.610 --> 00:01:26.680
So you divide both sides by 2.

00:01:26.680 --> 00:01:31.450
And you get x needs to be
greater than or equal to 4.

00:01:31.450 --> 00:01:36.070
So the domain here is the
set of all real numbers that

00:01:36.070 --> 00:01:37.879
are greater than
or equal to 4. x

00:01:37.879 --> 00:01:39.420
has to be greater
than or equal to 4.

00:01:39.420 --> 00:01:41.253
Or another way of saying
it is this function

00:01:41.253 --> 00:01:44.410
is defined when x is
greater than or equal to 4.

00:01:44.410 --> 00:01:45.730
And we're done.