SAT Prep: Test 1 Section 3 Part 5

WEBVTT
Kind: captions
Language: en

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Welcome back.

00:00:01.370 --> 00:00:03.030
Well we left off on
problem number 13.

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But let me start it over just so
that we don't get confused

00:00:07.170 --> 00:00:09.320
kind of doing the problem over
the course of two videos.

00:00:09.320 --> 00:00:10.680
So number 13.

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It starts off with
this pattern.

00:00:12.990 --> 00:00:20.070
3, 5, minus 5.

00:00:20.070 --> 00:00:25.330
And what it says in the problem
is that each even

00:00:25.330 --> 00:00:27.650
number term is 2 more than
the previous term.

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And then each odd number term
is negative 1 times the

00:00:30.580 --> 00:00:31.430
previous term.

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So we started with 3.

00:00:33.950 --> 00:00:41.400
So let me say term-- let me
write it here, term-- term and

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then value.

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And then what we did.

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Maybe we could call that
f of x or the function.

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How did we go from one
value to the next?

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So if we make a table,
term one is value 3.

00:01:01.080 --> 00:01:02.850
And that's just the
starting one.

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Term two is 5.

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That's because we added
2, we added 2 to 3.

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We could write 3 plus
2 is equal to 5.

00:01:16.840 --> 00:01:20.450
Term three is minus 5.

00:01:20.450 --> 00:01:29.070
And that's because minus 1 times
this 5-- let me draw

00:01:29.070 --> 00:01:32.500
these arrows-- is equal
to minus 5.

00:01:32.500 --> 00:01:34.920
And then we add again-- I'm
going to switch colors just to

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keep it interesting.

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So term four is going to be
minus 5 plus 2, which

00:01:47.220 --> 00:01:48.470
equals minus 3.

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Then term five is going to be
minus 3 times minus 1, which

00:02:01.380 --> 00:02:04.640
equals 3 again.

00:02:04.640 --> 00:02:05.180
Right?

00:02:05.180 --> 00:02:06.750
And I think you see how we're
getting these patterns.

00:02:06.750 --> 00:02:08.860
You just keep switching
between adding 2 and

00:02:08.860 --> 00:02:10.380
multiplying by negative 1.

00:02:10.380 --> 00:02:13.040
And then so term six is
going to be what?

00:02:13.040 --> 00:02:15.360
It's going to be this
plus 2, which is 5.

00:02:15.360 --> 00:02:16.650
I think you're seeing
the pattern again.

00:02:16.650 --> 00:02:18.960
Let me do a bunch more and
I'll do it outside of the

00:02:18.960 --> 00:02:20.150
scope of this chart.

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The term seven is going
to be what?

00:02:26.500 --> 00:02:29.180
It's going to be minus 1
times that, so minus 5.

00:02:29.180 --> 00:02:30.450
And you see the pattern now.

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Term eight is going
to be minus 3.

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Term nine is going to be what?

00:02:40.580 --> 00:02:42.540
Minus 3, minus 3.

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So here we added 2 so we
multiply it by negative 1.

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So it's going to be 3 again.

00:02:49.100 --> 00:02:50.540
And you could just experiment
with yourself.

00:02:50.540 --> 00:02:55.670
Once you understand how soon.
the sequence works, do a bunch

00:02:55.670 --> 00:02:58.370
of terms until you
see a cycle.

00:02:58.370 --> 00:03:00.130
And that's the key thing.

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In any SAT problem, when
they ask you like what

00:03:03.250 --> 00:03:05.800
is the 100th term?

00:03:05.800 --> 00:03:08.260
It's going to have a cycle
because they don't expect you

00:03:08.260 --> 00:03:10.000
to sit and write out 100
terms because you

00:03:10.000 --> 00:03:11.210
would run out of time.

00:03:11.210 --> 00:03:14.520
So if they ask you the 100th
term, the pattern probably

00:03:14.520 --> 00:03:19.030
repeats every three or four or
five or, at most, six terms.

00:03:19.030 --> 00:03:20.490
And then you can use that
to figure out what

00:03:20.490 --> 00:03:21.630
the 100th term is.

00:03:21.630 --> 00:03:23.780
So here we see that the pattern
repeats-- let's see,

00:03:23.780 --> 00:03:26.340
3, 5, negative 5, negative 3.

00:03:26.340 --> 00:03:29.590
3, 5, negative 5,
negative 3, 3.

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So we see it repeats every
fourth term, right?

00:03:31.900 --> 00:03:35.170
This is kind of one cycle, then
3, 5, negative 3, and

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this is another cycle.

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So it has this four pattern.

00:03:39.760 --> 00:03:45.950
So can we come up with any term
that is a multiple of 4,

00:03:45.950 --> 00:03:47.120
what is our value?

00:03:47.120 --> 00:03:52.070
Well at 4 our value
is minus 3.

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What's the next multiple of 4?

00:03:52.940 --> 00:03:54.160
8.

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At 8 our value's minus 3.

00:03:57.560 --> 00:04:00.780
So in general-- and if we did
12 we'd also get minus 3.

00:04:00.780 --> 00:04:03.070
If you don't believe me
you should try it out.

00:04:03.070 --> 00:04:07.730
So at any multiple of 4, we
know our value is minus 3.

00:04:07.730 --> 00:04:16.500
So what I do is, they want
to know the 55th term.

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So what term is a multiple of
4 that's closest to 55?

00:04:22.750 --> 00:04:25.610
Let's see, is 56 a
multiple of 4?

00:04:25.610 --> 00:04:25.800
Yeah.

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56-- that's 10.

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14 times 4 is 56, right?

00:04:31.330 --> 00:04:36.800
So the 56th term, we know the
56th term because 56 is a

00:04:36.800 --> 00:04:39.940
multiple of 4, just like
4 and 8 and 12.

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So the 56th term-- I have
trouble saying that.

00:04:43.980 --> 00:04:45.175
Maybe I have a little lisp.

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56th term is negative
3, right?

00:04:47.540 --> 00:04:49.640
Because it's a multiple of 4.

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If the 56th term is negative
3, what is the 55th term?

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Well the term before the
negative 3 term is always the

00:04:59.260 --> 00:05:00.700
negative 5 term, right?

00:05:00.700 --> 00:05:01.830
Up here, negative 3?

00:05:01.830 --> 00:05:03.820
It's always negative 5.

00:05:03.820 --> 00:05:08.880
So the answer is negative
5 is the 55th term, and

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that is choice A.

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Very good.

00:05:11.780 --> 00:05:13.685
All right, let's move on
to the next question.

00:05:18.410 --> 00:05:20.690
I know that one's a little hard,
and those you always

00:05:20.690 --> 00:05:23.630
just have to experiment a little
bit, try to see the

00:05:23.630 --> 00:05:26.150
cycle, and then make sure you
have your terms right.

00:05:26.150 --> 00:05:27.860
You know that you're starting
on the first term as opposed

00:05:27.860 --> 00:05:28.820
to the 0 term.

00:05:28.820 --> 00:05:32.190
And then figure out what the
easiest value is as close what

00:05:32.190 --> 00:05:34.080
they're asking as possible and
then you can kind of move

00:05:34.080 --> 00:05:35.310
backwards or forwards.

00:05:35.310 --> 00:05:36.180
That's how I do it.

00:05:36.180 --> 00:05:39.120
That's so you don't get too
confused with the shifts on

00:05:39.120 --> 00:05:40.320
the terms.

00:05:40.320 --> 00:05:42.300
All right, number 14.

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In the xy plane, the equation
you of line l is-- let me

00:05:46.190 --> 00:05:48.910
write that down-- number 14.

00:05:48.910 --> 00:05:56.730
The equation of line l is
y is equal to 2x plus 5.

00:05:56.730 --> 00:06:01.920
If the line m is the reflection
of l in the x-axis,

00:06:01.920 --> 00:06:03.980
what is the equation
of line m?

00:06:03.980 --> 00:06:05.570
So I'm going to draw a graph
for you, just so you

00:06:05.570 --> 00:06:06.540
understand what I'm doing.

00:06:06.540 --> 00:06:08.690
But on the real SAT you really
don't have to do this because

00:06:08.690 --> 00:06:10.120
you wouldn't want to
run out of time.

00:06:16.400 --> 00:06:18.610
Oh man, I'm going to run out
of time if I can't draw a

00:06:18.610 --> 00:06:21.430
straight line.

00:06:21.430 --> 00:06:23.770
I'm being a stickler for this.

00:06:23.770 --> 00:06:25.260
Oh, I still couldn't do it.

00:06:25.260 --> 00:06:26.510
I'll move forward.

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So this line, so y
equals 2x plus 5.

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So if you remember from our
slope and y-intercept modules,

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the y-intercept's going
to be 5, right here.

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And then the line's going
to have a slope of 2.

00:06:48.530 --> 00:06:49.330
Right?

00:06:49.330 --> 00:06:51.253
So the line's going to look
something like this.

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That's line l.

00:07:01.770 --> 00:07:03.890
This is line l.

00:07:03.890 --> 00:07:07.740
And then we want to know, they
say line m is the reflection

00:07:07.740 --> 00:07:09.240
of l in the x-axis.

00:07:09.240 --> 00:07:10.280
So reflection would
kind of be the

00:07:10.280 --> 00:07:12.540
x-axis-- this is the x-axis.

00:07:12.540 --> 00:07:14.860
So it's like if we were looking
at this in a pool of

00:07:14.860 --> 00:07:18.380
water, the reflection would be
like if you took this and

00:07:18.380 --> 00:07:20.180
flipped it over the x-axis.

00:07:20.180 --> 00:07:23.120
So the line m is going to look
something like this.

00:07:27.440 --> 00:07:29.650
Line m is going to
look like this.

00:07:32.950 --> 00:07:33.570
So a couple of things.

00:07:33.570 --> 00:07:35.060
It's like the mirror image.

00:07:35.060 --> 00:07:40.920
If we flipped it, it's like we
flipped it over the axis.

00:07:40.920 --> 00:07:44.210
So if the y-intercept here is
5, the y-intercept here is

00:07:44.210 --> 00:07:46.076
going to be minus 5.

00:07:46.076 --> 00:07:46.680
Right?

00:07:46.680 --> 00:07:47.870
Because we just flipped it.

00:07:47.870 --> 00:07:53.210
And similarly, if every one
this moves up 2, right,

00:07:53.210 --> 00:07:54.110
because that's at slope.

00:07:54.110 --> 00:07:54.670
[UNINTELLIGIBLE]

00:07:54.670 --> 00:07:55.280
2.

00:07:55.280 --> 00:07:57.340
For every one we move here
to the right we're

00:07:57.340 --> 00:07:58.830
going to go down 2.

00:07:58.830 --> 00:08:00.670
So the slope is going to be
the negative of this or

00:08:00.670 --> 00:08:02.240
negative 2.

00:08:02.240 --> 00:08:07.165
So line m, so this is line m,
the equation of line m's going

00:08:07.165 --> 00:08:09.480
to be y is equal to-- it's going
to have the negative

00:08:09.480 --> 00:08:10.500
slope, right?

00:08:10.500 --> 00:08:12.140
The same slope, but
it just moves in

00:08:12.140 --> 00:08:13.100
the downward direction.

00:08:13.100 --> 00:08:15.220
So it's going to be minus 2x.

00:08:15.220 --> 00:08:19.150
And then its y-intercept is--
instead of plus 5 it's going

00:08:19.150 --> 00:08:20.320
to be minus 5.

00:08:20.320 --> 00:08:23.130
Minus 2x minus 5.

00:08:23.130 --> 00:08:26.990
And that is choice A.

00:08:26.990 --> 00:08:29.190
Looks like the next problem
involves a lot of drawing, and

00:08:29.190 --> 00:08:31.440
I only have a minute and a
half left in this video.

00:08:31.440 --> 00:08:35.330
So I will do the next problem
in the next video.

00:08:35.330 --> 00:08:36.950
I will see


SAT Prep: Test 1 Section 3 Part 4

WEBVTT
Kind: captions
Language: en

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Welcome back.

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Well, we're now on problem
number 11.

00:00:06.250 --> 00:00:11.040
And this says the circle above
has center o, and diameter ab.

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And it's a dotted line in the
book, but I made it a solid

00:00:14.220 --> 00:00:15.190
green line.

00:00:15.190 --> 00:00:18.500
The two semi-circles have
diameters oa, so this is one

00:00:18.500 --> 00:00:21.600
semi-circle and this is
its diameter, oa.

00:00:21.600 --> 00:00:24.370
And then ob, this is the
other semi-circle.

00:00:24.370 --> 00:00:25.500
It's ob.

00:00:25.500 --> 00:00:28.870
If the circumference of
the circle is 36 pi.

00:00:28.870 --> 00:00:32.020
So, circumference of the circle,
of the big circle--

00:00:32.020 --> 00:00:34.250
and actually I'll do it in
yellow, which is the color of

00:00:34.250 --> 00:00:37.840
kind of the perimeter or the
circumference of that circle.

00:00:37.840 --> 00:00:42.950
So they're saying the
circumference is 36 pi.

00:00:42.950 --> 00:00:46.090
What is the length of the curved
path from ab to o?

00:00:46.090 --> 00:00:49.680
So they want to figure out the
length of, essentially, this

00:00:49.680 --> 00:00:51.260
brown path.

00:00:51.260 --> 00:00:53.210
And so before we break into
the math, just how can we

00:00:53.210 --> 00:00:54.540
think about this?

00:00:54.540 --> 00:00:58.580
Well we have a big circle and
we know its circumference.

00:00:58.580 --> 00:01:00.850
So if we know its circumference
we could figure

00:01:00.850 --> 00:01:02.730
out its radius or its
diameter, right?

00:01:02.730 --> 00:01:06.860
Because you know the formula,
circumference is

00:01:06.860 --> 00:01:09.820
equal to 2 pi r.

00:01:09.820 --> 00:01:12.070
So if we know the circumference
of this big

00:01:12.070 --> 00:01:14.540
circle, we can actually
figure out its radius

00:01:14.540 --> 00:01:16.560
by using this formula.

00:01:16.560 --> 00:01:18.960
And the radius of the big
circle, which would be-- let's

00:01:18.960 --> 00:01:22.670
say, oa or ob-- they're both
radiuses of the big circle--

00:01:22.670 --> 00:01:27.540
than those are the diameter of
each of these half circles or

00:01:27.540 --> 00:01:29.810
each of these semi-circles.

00:01:29.810 --> 00:01:34.820
So if we know this diameter we
could take half of it and

00:01:34.820 --> 00:01:38.880
re-use this formula, or say the
circumference is pi times

00:01:38.880 --> 00:01:39.370
the diameter.

00:01:39.370 --> 00:01:40.780
We could do either one.

00:01:40.780 --> 00:01:45.700
And we could figure out-- let's
say we know this radius,

00:01:45.700 --> 00:01:47.730
so we could figure out this.

00:01:47.730 --> 00:01:49.020
Then we could figure out
the circumference

00:01:49.020 --> 00:01:51.410
of this larger circle.

00:01:51.410 --> 00:01:53.370
And I'll actually make the claim
that the circumference

00:01:53.370 --> 00:01:57.360
of this larger circle is the
same as the circumference of

00:01:57.360 --> 00:01:59.232
this half circle plus
this half circle.

00:01:59.232 --> 00:02:00.770
And I think you know why.

00:02:00.770 --> 00:02:04.340
If you just shifted this up then
you would get the other

00:02:04.340 --> 00:02:06.460
half of this circle.

00:02:06.460 --> 00:02:07.940
The other way you could have
done it is, you figure out the

00:02:07.940 --> 00:02:10.900
circumference of this circle,
divide by 2 to get this.

00:02:10.900 --> 00:02:13.840
Take the circumference of this
circle, divide by 2 to get

00:02:13.840 --> 00:02:14.740
this and then add them.

00:02:14.740 --> 00:02:16.800
But you'll get the same thing
as just the circumference of

00:02:16.800 --> 00:02:17.890
this circle.

00:02:17.890 --> 00:02:19.540
So let's do the problem.

00:02:19.540 --> 00:02:22.240
So circumference is
equal to 2 pi r.

00:02:22.240 --> 00:02:25.420
We know the circumference of
the larger circle is 36 pi.

00:02:25.420 --> 00:02:32.930
So let's see, 36 pi is
equal to 2 pi r.

00:02:32.930 --> 00:02:36.490
Divide both sides by 2 pi, and
I'll switch the sides too.

00:02:36.490 --> 00:02:42.565
So you get r is equal to 36 pi
over 2 pi-- I just divided

00:02:42.565 --> 00:02:44.020
both sides by 2 pi.

00:02:44.020 --> 00:02:45.520
pi's cancel out.

00:02:45.520 --> 00:02:51.200
36 divided by 2 is
equal to 18.

00:02:51.200 --> 00:02:52.220
And so what's 18?

00:02:52.220 --> 00:02:58.340
18 is this length right here.

00:02:58.340 --> 00:03:00.090
Is this length.

00:03:00.090 --> 00:03:01.360
We just figured out,
that's the radius

00:03:01.360 --> 00:03:02.880
of the larger circle.

00:03:02.880 --> 00:03:05.340
And now we could use this, which
is the diameter of the

00:03:05.340 --> 00:03:08.020
smaller circle to figure out
its circumference using the

00:03:08.020 --> 00:03:08.890
same formula.

00:03:08.890 --> 00:03:12.400
And now I'm going to use brown
for this circumference.

00:03:12.400 --> 00:03:15.870
So circumference is
equal to 2 pi r.

00:03:15.870 --> 00:03:18.410
And that's the same thing as--
because 2 times the radius is

00:03:18.410 --> 00:03:19.330
the diameter.

00:03:19.330 --> 00:03:22.040
So that equals pi times
the diameter.

00:03:22.040 --> 00:03:26.760
We know what the diameter
is, the diameter is 18.

00:03:26.760 --> 00:03:29.260
So we could say circumference
is equal to

00:03:29.260 --> 00:03:30.880
pi times the diameter.

00:03:30.880 --> 00:03:35.690
The diameter is 18.

00:03:35.690 --> 00:03:38.120
Or if we rearrange that
that's 18 pi.

00:03:38.120 --> 00:03:39.030
And we're done.

00:03:39.030 --> 00:03:41.450
Because what we just figured
out is the circumference of

00:03:41.450 --> 00:03:43.360
this larger circle.

00:03:43.360 --> 00:03:47.760
We could divide it by 2 and
figure out-- we could say that

00:03:47.760 --> 00:03:50.530
this is 9 pi right here.

00:03:50.530 --> 00:03:51.560
Just this piece.

00:03:51.560 --> 00:03:52.710
This piece is 9 pi.

00:03:52.710 --> 00:03:55.250
And if we do the same math here
on this piece, you would

00:03:55.250 --> 00:03:57.090
also get 9 pi.

00:03:57.090 --> 00:03:59.710
But that also adds
up to 18 pi.

00:03:59.710 --> 00:04:02.570
Hopefully you understand
what I'm saying.

00:04:02.570 --> 00:04:04.250
Let's move on to the
next problem.

00:04:09.370 --> 00:04:14.760
So in the table above-- this is
our classic f of x table.

00:04:17.700 --> 00:04:21.000
When x is 0, f of x is a.

00:04:21.000 --> 00:04:25.020
When x is 1, f of x is 24.

00:04:25.020 --> 00:04:28.700
When x is 2, f of x is b.

00:04:28.700 --> 00:04:30.840
It says the table above shows
some values of the functions

00:04:30.840 --> 00:04:33.100
of f. f is a linear function.

00:04:33.100 --> 00:04:35.130
So what does a linear
function mean?

00:04:35.130 --> 00:04:37.710
Linear means it's a line.

00:04:37.710 --> 00:04:41.200
And that also means that
the slope is constant.

00:04:41.200 --> 00:04:43.960
If you haven't gottem-- well,
if you haven't got past

00:04:43.960 --> 00:04:46.120
algebra I, you probably haven't
seen functions with

00:04:46.120 --> 00:04:47.220
non-constant slopes.

00:04:47.220 --> 00:04:48.780
But like, for example,
quadratics don't

00:04:48.780 --> 00:04:50.030
have constant slopes.

00:04:50.030 --> 00:04:51.845
The linear means that the
slope is constant.

00:04:54.950 --> 00:04:58.870
That's the one conclusion you
can make from that statement.

00:04:58.870 --> 00:05:03.760
And then they want to know, what
is the value of a plus b?

00:05:03.760 --> 00:05:05.530
a plus b is what?

00:05:05.530 --> 00:05:07.800
So the only thing we know is
this table and we know that

00:05:07.800 --> 00:05:08.910
it's linear.

00:05:08.910 --> 00:05:12.560
And linear means that the
slope is constant.

00:05:12.560 --> 00:05:14.040
So the slope is constant.

00:05:14.040 --> 00:05:16.040
So let's see what we can
figure out the slope.

00:05:16.040 --> 00:05:19.860
So that means the slope from x
is equal to 0 to x is equal to

00:05:19.860 --> 00:05:23.210
1 is the same as the slope from
x is equal to 1, to x is

00:05:23.210 --> 00:05:24.480
equal to 2.

00:05:24.480 --> 00:05:28.140
So what's the slope if we
use these two terms?

00:05:28.140 --> 00:05:31.500
Slope is change in-- well change
in f of x over change

00:05:31.500 --> 00:05:32.780
in x, right?

00:05:32.780 --> 00:05:36.110
So change in-- I mean it's
change in y over

00:05:36.110 --> 00:05:37.250
change in x, right?

00:05:37.250 --> 00:05:40.750
We could call this y instead.

00:05:40.750 --> 00:05:45.730
So change in y-- let's see, 24
minus a-- and you might want

00:05:45.730 --> 00:05:48.680
to review the slope video
if you forgot this.

00:05:48.680 --> 00:05:53.360
24 minus a over the change
in x is 1 minus 0.

00:05:53.360 --> 00:05:55.100
Well that's just 1.

00:05:55.100 --> 00:05:58.800
So that equals 24
minus a, right?

00:05:58.800 --> 00:06:03.310
And what's the slope
from these points?

00:06:03.310 --> 00:06:04.080
Because we know it's constant.

00:06:04.080 --> 00:06:06.810
We know the slope between these
two points is going to

00:06:06.810 --> 00:06:08.930
be the same as the slope between
these two points.

00:06:08.930 --> 00:06:09.900
Well, we do the same thing.

00:06:09.900 --> 00:06:17.880
We take the end points' change
in y, so it's b minus 24 over

00:06:17.880 --> 00:06:20.770
change in x, 2 minus 1.

00:06:20.770 --> 00:06:23.240
So that equals-- well that's
just 1, so that

00:06:23.240 --> 00:06:25.660
equals b minus 24.

00:06:25.660 --> 00:06:29.490
So this is the slope between
these second two points, and

00:06:29.490 --> 00:06:31.600
this the slope between
the first two points.

00:06:31.600 --> 00:06:35.130
And since we know it's linear,
since we know this is a linear

00:06:35.130 --> 00:06:38.180
equation, we can say well they
must be equal to each other.

00:06:38.180 --> 00:06:40.630
So let's see if that gets us
to where we want to go.

00:06:40.630 --> 00:06:53.760
So we know that 24 minus a
is equal to b minus 24.

00:06:53.760 --> 00:06:57.110
Let's add 24 to both sides.

00:06:57.110 --> 00:07:01.760
You get 48 minus a
is equal to b.

00:07:01.760 --> 00:07:03.600
I just added 24 to both sides.

00:07:03.600 --> 00:07:08.110
And if we add a to both sides,
I see the end point-- 48 is

00:07:08.110 --> 00:07:10.980
equal to a plus b.

00:07:10.980 --> 00:07:14.570
And that is choice C.

00:07:14.570 --> 00:07:16.110
Let us move on.

00:07:16.110 --> 00:07:18.430
And remember, all we did is we
used this information, and we

00:07:18.430 --> 00:07:20.160
used the fact that slope
is constant.

00:07:20.160 --> 00:07:21.670
So the only thing that you would
have had to know ahead

00:07:21.670 --> 00:07:26.060
of time is the equation of the
slope, and that's change in y

00:07:26.060 --> 00:07:29.260
over change in x, and that
linear means constant slope.

00:07:35.550 --> 00:07:37.990
So they write a little
sequence up here,

00:07:37.990 --> 00:07:41.500
this is number 13.

00:07:41.500 --> 00:07:46.500
It's 3, 5, negative 5.

00:07:46.500 --> 00:07:48.540
They say the first term in the
sequence of the numbers shown

00:07:48.540 --> 00:07:49.720
is 3-- that's obvious.

00:07:49.720 --> 00:07:50.280
They show that.

00:07:50.280 --> 00:07:54.880
Each even number term is 2 more
than the previous term,

00:07:54.880 --> 00:07:58.960
so we go from 3 to 5, that was
2 more, because this is the

00:07:58.960 --> 00:08:00.630
second term, so it's an
even number term.

00:08:00.630 --> 00:08:02.660
So this is 2 more.

00:08:02.660 --> 00:08:06.350
And each odd number of term
after the first is minus 1

00:08:06.350 --> 00:08:07.330
times the previous.

00:08:07.330 --> 00:08:11.160
So minus 1 times this
term is minus 5.

00:08:11.160 --> 00:08:12.610
So that's what they're saying.

00:08:12.610 --> 00:08:15.580
For example, the second term
is 3 plus 2, and the third

00:08:15.580 --> 00:08:17.640
term is minus 1 times 5.

00:08:17.640 --> 00:08:20.240
What is the 55th term
of the sequence?

00:08:20.240 --> 00:08:23.400
So on the SAT, whenever they ask
you some crazy term, like

00:08:23.400 --> 00:08:27.250
the 100th term or the 55th term,
that a huge signal that

00:08:27.250 --> 00:08:29.820
this sequence is probably going
to repeat itself and you

00:08:29.820 --> 00:08:31.010
just have to figure
out the pattern.

00:08:31.010 --> 00:08:32.450
And then once you know that
pattern I'll show you how to

00:08:32.450 --> 00:08:34.710
figure out the 500th term
if you have to.

00:08:34.710 --> 00:08:37.789
So with any sequence the best
thing to do is just write out

00:08:37.789 --> 00:08:43.210
a bunch of terms. So we start
at 3, we add 2, we get 5.

00:08:43.210 --> 00:08:45.660
We multiply that
by negative 1.

00:08:45.660 --> 00:08:47.145
This is plus 2.

00:08:47.145 --> 00:08:51.450
So we multiply by negative
1, times negative 1,

00:08:51.450 --> 00:08:53.130
we get minus 5.

00:08:53.130 --> 00:08:54.970
Then we want to add 2.

00:08:54.970 --> 00:09:01.350
So 2 times negative 5 is--
we're adding 2, right.

00:09:01.350 --> 00:09:05.080
2 plus negative 5 is minus 3.

00:09:05.080 --> 00:09:07.320
I think I'm already seeing
a pattern here.

00:09:07.320 --> 00:09:09.500
And then we have to multiply
that times minus

00:09:09.500 --> 00:09:10.430
1 times minus 1.

00:09:10.430 --> 00:09:12.970
That's not x minus 1, this
is times minus 1.

00:09:12.970 --> 00:09:15.640
So minus 1 times 3.

00:09:15.640 --> 00:09:19.630
So minus 1 times 3-- I should
do minus 1 time instead.

00:09:19.630 --> 00:09:23.460
Minus 1 times 3,
well that's 3.

00:09:23.460 --> 00:09:28.310
And then we add 2 to it,
plus 2, we get 5.

00:09:28.310 --> 00:09:30.120
And it's interesting, then it's
going to start repeating.

00:09:30.120 --> 00:09:32.610
Minus 1 is minus 5.

00:09:32.610 --> 00:09:35.660
So let's see what the repetition
looks like.

00:09:35.660 --> 00:09:40.390
The first term is 3, the second
term is 5, the third

00:09:40.390 --> 00:09:46.000
term is minus 5, the fourth
term is minus 3.

00:09:46.000 --> 00:09:49.770
The fifth term is 3 again.

00:09:49.770 --> 00:09:53.010
The sixth term is
5, 7 and so on.

00:09:53.010 --> 00:09:53.930
I'm almost out of time.

00:09:53.930 --> 00:09:56.130
I'll continue this in
the next video.


SAT Prep: Test 1 Section 3 Part 3

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:01.500
Welcome back.

00:00:01.500 --> 00:00:04.660
Before I move onto the next
problem, sometimes I rush near

00:00:04.660 --> 00:00:06.290
the end of the video just
because I want to get under

00:00:06.290 --> 00:00:08.570
the YouTube 10-minute limit.

00:00:08.570 --> 00:00:12.060
But I just want to make sure you
understand the conclusions

00:00:12.060 --> 00:00:13.570
I reached in this problem
because it's kind of

00:00:13.570 --> 00:00:14.820
interesting.

00:00:14.820 --> 00:00:17.950
If I have a bunch of numbers and
when I multiply them I get

00:00:17.950 --> 00:00:21.320
a non-zero number, I know none
of these numbers can be 0,

00:00:21.320 --> 00:00:24.350
because as you know, if any
one of them was 0 or maybe

00:00:24.350 --> 00:00:27.390
more than one, then this
value would be 0.

00:00:27.390 --> 00:00:31.160
So we know none of these
can be 0, rst or v.

00:00:31.160 --> 00:00:35.580
And similarly, between stu and
v-- let me re-write that, stu

00:00:35.580 --> 00:00:39.930
and v equals 0-- we know that at
least one of these must be

00:00:39.930 --> 00:00:42.770
0 because when you multiply
it out you get 0.

00:00:42.770 --> 00:00:43.990
Maybe all of them are.

00:00:43.990 --> 00:00:47.100
Well we know s is not 0 because
it was up here.

00:00:47.100 --> 00:00:49.590
We knew t was not 0 and we knew
v were not 0 because they

00:00:49.590 --> 00:00:51.200
were all up on this top one.

00:00:51.200 --> 00:00:53.310
And if any of them were
0, this top equation

00:00:53.310 --> 00:00:54.110
would have been 0.

00:00:54.110 --> 00:00:56.170
And that's how we reached
that conclusion.

00:00:56.170 --> 00:00:58.040
So let's move on.

00:00:58.040 --> 00:01:00.970
I hope I'm not confusing you.

00:01:00.970 --> 00:01:03.560
I'm doing it with a bit of a
sense of urgency just because,

00:01:03.560 --> 00:01:07.600
well I guess one should
when taking the SAT.

00:01:07.600 --> 00:01:09.100
All right, number 8.

00:01:09.100 --> 00:01:16.060
During a game the blue team
scored 1/6 of its-- so let me

00:01:16.060 --> 00:01:17.430
do the blue team in blue.

00:01:20.180 --> 00:01:23.800
The blue team scored
1/6 of its points

00:01:23.800 --> 00:01:25.540
in the first quarter.

00:01:25.540 --> 00:01:31.560
So Q1 they scored
1/6 of points.

00:01:31.560 --> 00:01:33.580
1/4 in the second quarter.

00:01:33.580 --> 00:01:37.370
So in Q2 scored 1/4
of its points.

00:01:40.570 --> 00:01:42.620
1/3 in the third quarter.

00:01:42.620 --> 00:01:47.310
So in Q3 they scored
1/3 of its points.

00:01:47.310 --> 00:01:50.840
1/3 in the third quarter, and
remaining points in the fourth

00:01:50.840 --> 00:01:53.010
quarter, which makes sense.

00:01:57.050 --> 00:02:11.310
If the total score was 36, how
many points did the blue team

00:02:11.310 --> 00:02:14.830
score in the fourth quarter?

00:02:14.830 --> 00:02:18.440
Well the total points
are 36, right?

00:02:18.440 --> 00:02:21.490
So it says they scored 1/6
in the first quarter.

00:02:21.490 --> 00:02:25.660
So 1/6 of 36-- you could do that
in your head or you could

00:02:25.660 --> 00:02:26.915
just multiply fractions.

00:02:26.915 --> 00:02:30.070
If you say 1/6 times 36.

00:02:30.070 --> 00:02:33.440
That equals 36 over
6, which equals 6.

00:02:33.440 --> 00:02:36.010
So they scored 6 points
in the first quarter.

00:02:36.010 --> 00:02:37.430
How many did they score
in the second quarter?

00:02:37.430 --> 00:02:41.580
Well what's 1/4 of 36.

00:02:41.580 --> 00:02:45.900
So you just multiply 1/4
times 36 and you get 9.

00:02:45.900 --> 00:02:48.840
1/3 of 36, well that's 12.

00:02:48.840 --> 00:02:52.010
So in the first three quarters
we scored what?

00:02:52.010 --> 00:02:54.720
6 plus 9 is 15.

00:02:54.720 --> 00:02:57.750
15 plus 12 is 27.

00:02:57.750 --> 00:03:01.880
So we scored 27 in the first
three quarters, right?

00:03:01.880 --> 00:03:04.130
And then we scored
the rest in Q4.

00:03:04.130 --> 00:03:08.030
So it would be the total points
minus what we scored in

00:03:08.030 --> 00:03:11.430
the first three quarters,
which is 27.

00:03:11.430 --> 00:03:16.030
And 36 minus 27 is 9 points.

00:03:16.030 --> 00:03:19.070
And if I look at the choices,
that is choice C.

00:03:19.070 --> 00:03:22.675
And if I made a mistake it's
because I cannot add, subtract

00:03:22.675 --> 00:03:23.870
or multiply fractions.

00:03:23.870 --> 00:03:25.910
But that's all there
is to this problem.

00:03:25.910 --> 00:03:27.160
Let's move on.

00:03:29.850 --> 00:03:31.690
Sometimes I feel like I'm going
too fast, so I want to

00:03:31.690 --> 00:03:35.170
make sure I'm not
confusing you.

00:03:39.710 --> 00:03:40.970
I like these type because
I don't have to

00:03:40.970 --> 00:03:41.990
write a lot or draw.

00:03:41.990 --> 00:03:44.420
Well I like the drawing,
it just takes time.

00:03:44.420 --> 00:03:46.470
I will switch colors to keep
you from sleeping.

00:03:49.060 --> 00:03:53.570
2 to the 2x-- this is
problem number 9.

00:03:57.010 --> 00:04:05.600
2 to the 2x is equal to
8 to the x minus 1.

00:04:05.600 --> 00:04:08.070
They just want to
know, what is x?

00:04:08.070 --> 00:04:09.720
So at first you might
be confused.

00:04:09.720 --> 00:04:12.980
But the trick to doing these
kind of exponential-- I guess

00:04:12.980 --> 00:04:15.010
we can call them an exponential
equation-- is to

00:04:15.010 --> 00:04:16.290
get our bases the same.

00:04:16.290 --> 00:04:18.390
And once we get the bases the
same then we can set the

00:04:18.390 --> 00:04:19.680
exponents equal to each other.

00:04:19.680 --> 00:04:21.430
So how do I do that?

00:04:21.430 --> 00:04:24.830
Well can I write 8
as a power of 2?

00:04:24.830 --> 00:04:25.840
Well sure, what's 8?

00:04:25.840 --> 00:04:27.790
It's 2 to the third power.

00:04:27.790 --> 00:04:29.840
So let's rewrite 8 as 2
to the third power.

00:04:29.840 --> 00:04:31.430
So let's figure out
the whole thing.

00:04:31.430 --> 00:04:38.460
2 to the 2x is equal to--
instead of 8, I'm going to

00:04:38.460 --> 00:04:40.050
write 2 to the third power.

00:04:40.050 --> 00:04:41.690
Because that's all 8 is.

00:04:41.690 --> 00:04:46.860
And hopefully you can review
the exponent video if you

00:04:46.860 --> 00:04:47.920
don't remember that.

00:04:47.920 --> 00:04:53.110
And this 8 is this, and then
that's going to the x minus 1.

00:04:53.110 --> 00:04:54.950
And then if you remember your
exponent rules, and you should

00:04:54.950 --> 00:04:57.600
remember your exponent rules
before taking the SAT, you

00:04:57.600 --> 00:04:59.830
know that if you raise something
to a power and then

00:04:59.830 --> 00:05:01.950
raise that to a power,
you can just

00:05:01.950 --> 00:05:03.850
multiply these two exponents.

00:05:03.850 --> 00:05:12.170
So that gets us to 2 to the 2x
is equal to 2 to the 3 times x

00:05:12.170 --> 00:05:14.780
minus 1, right, and you
distribute the 3.

00:05:14.780 --> 00:05:19.040
So it's 2 to the 3x minus 3.

00:05:19.040 --> 00:05:20.820
3 times x, 3 times minus 1.

00:05:20.820 --> 00:05:22.230
3x minus 3.

00:05:22.230 --> 00:05:26.700
Now 2 to the 2x is equal to 2 to
the 3x minus 3, so we know

00:05:26.700 --> 00:05:30.480
that the two exponents must be
equal to each other because we

00:05:30.480 --> 00:05:31.900
have the same base.

00:05:31.900 --> 00:05:34.560
Now this becomes a level one
linear equations problem.

00:05:34.560 --> 00:05:39.660
2x is equal to 3x minus 3.

00:05:39.660 --> 00:05:43.650
We can subtract 3x from both
sides and we get minus x is

00:05:43.650 --> 00:05:49.530
equal to minus 3, or
x is equal to 3.

00:05:49.530 --> 00:05:50.650
And we are done.

00:05:50.650 --> 00:05:51.900
And this is choice number B.

00:05:54.580 --> 00:06:05.780
Turning the page, number 10.

00:06:05.780 --> 00:06:10.460
If 4 less-- I like these-- If 4
less than 3 times a certain

00:06:10.460 --> 00:06:14.990
number is 2 more than the
number, what is the number?

00:06:14.990 --> 00:06:19.080
So this is literally turning
English into math.

00:06:19.080 --> 00:06:21.010
So I'm going to actually
write the whole thing.

00:06:21.010 --> 00:06:22.680
Not that you should do this on
the SAT because it'll take you

00:06:22.680 --> 00:06:23.660
too much time.

00:06:23.660 --> 00:06:26.210
But you can do this kind of
with their written word.

00:06:26.210 --> 00:06:55.400
If 4 less than 3 times a certain
number is 2 more than

00:06:55.400 --> 00:07:00.870
the number, what
is the number?

00:07:08.150 --> 00:07:11.540
So instead of a certain number I
like to replace that with x,

00:07:11.540 --> 00:07:13.390
just because I'm going to
have to solve for x.

00:07:13.390 --> 00:07:15.135
So I'm going to change
this and I'm going

00:07:15.135 --> 00:07:17.100
to say this is x.

00:07:17.100 --> 00:07:22.700
What is x-- more than
the number x.

00:07:22.700 --> 00:07:27.740
So now we have if 4 less than
3 times x is 2 more than x,

00:07:27.740 --> 00:07:29.130
what is the number?

00:07:29.130 --> 00:07:31.795
So what is 4 less
than 3 times x?

00:07:34.630 --> 00:07:37.040
Well 3 times x is what?

00:07:37.040 --> 00:07:40.040
It's 3x.

00:07:40.040 --> 00:07:48.915
3 times x-- so if I took this
part, 3 times x, that's 3x.

00:07:48.915 --> 00:07:57.330
And then 4 less than that
is going to be minus 4.

00:07:57.330 --> 00:08:01.240
And that is, and I view is-- you
could view that as equals.

00:08:01.240 --> 00:08:03.180
You could have replaced the
word equals with is and it

00:08:03.180 --> 00:08:06.170
wouldn't have changed the
meaning of the sentence.

00:08:06.170 --> 00:08:12.670
So that equals 2 more than the
number, or in our case, x.

00:08:12.670 --> 00:08:14.890
So what is 2 more than x?

00:08:14.890 --> 00:08:16.480
It's x plus 2.

00:08:16.480 --> 00:08:19.830
2 more than x, you could just
rewrite in math as x plus 2.

00:08:19.830 --> 00:08:22.540
And then we solve
the equation.

00:08:22.540 --> 00:08:27.590
So let's see, you could add 4
to both sides, you get 3x is

00:08:27.590 --> 00:08:30.040
equal to x plus 6.

00:08:30.040 --> 00:08:31.620
Subtract x from both sides.

00:08:31.620 --> 00:08:34.594
You get 2x is equal to 6.

00:08:34.594 --> 00:08:41.130
And divide both sides by 2, and
you get x is equal to 3.

00:08:41.130 --> 00:08:44.610
And there we go,
it's choice E.

00:08:44.610 --> 00:08:46.470
I will stop this video
here because the next

00:08:46.470 --> 00:08:47.250
one involves drawing.

00:08:47.250 --> 00:08:49.020
I don't want to draw
on your time.

00:08:49.020 --> 00:08:50.860
I'll see you soon.


SAT Prep: Test 1 Section 3 Part 2

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:01.660
Welcome back.

00:00:01.660 --> 00:00:03.180
Well we're now on number 4.

00:00:03.180 --> 00:00:06.000
And just so you know how
considerate I am of your time,

00:00:06.000 --> 00:00:10.510
I drew this very fancy graphic
before the video just so not

00:00:10.510 --> 00:00:12.790
to waste your valuable time.

00:00:12.790 --> 00:00:13.970
So let's read the question.

00:00:13.970 --> 00:00:18.270
This is the graph that they--
and this X right here is

00:00:18.270 --> 00:00:19.520
number of children.

00:00:22.380 --> 00:00:24.280
And these are years.

00:00:24.280 --> 00:00:28.120
So the question says, the graph
above shows how the

00:00:28.120 --> 00:00:34.270
Jackson family continued to grow
between 1991 and 1998 by

00:00:34.270 --> 00:00:36.980
indicating the total number of
children in the family at the

00:00:36.980 --> 00:00:38.450
end of each year.

00:00:38.450 --> 00:00:42.070
The Jackson's have one set of
twins who were born to Miss

00:00:42.070 --> 00:00:45.460
Jackson one year-- because they
couldn't have been born

00:00:45.460 --> 00:00:47.850
to Mr. Jackson-- who
are born to Miss

00:00:47.850 --> 00:00:49.700
Jackson one year in July.

00:00:49.700 --> 00:00:52.170
During what year were
the twins born?

00:00:52.170 --> 00:00:55.020
Well let's see, this graph, you
see the years, each year

00:00:55.020 --> 00:00:57.320
they say how many children
are in the family.

00:00:57.320 --> 00:01:00.850
So we can figure out how many
children are born in each year

00:01:00.850 --> 00:01:05.220
just by figuring out how much
does the graph move up.

00:01:05.220 --> 00:01:10.450
For example, there were-- let
me say children born.

00:01:10.450 --> 00:01:14.020
And you could do this in your
head during the exam because

00:01:14.020 --> 00:01:16.270
you have to do it really, really
fast. But I'll explain

00:01:16.270 --> 00:01:18.510
it a little bit more basically
so that you can make

00:01:18.510 --> 00:01:19.680
sure you get it.

00:01:19.680 --> 00:01:21.610
So in 1991 were any
children born?

00:01:21.610 --> 00:01:24.290
Well no, because they
had no children.

00:01:24.290 --> 00:01:26.970
In '92 they went from
having 0 children in

00:01:26.970 --> 00:01:28.520
'91 to having 1 child.

00:01:28.520 --> 00:01:32.150
So the 1 child was born in
'92, and then how many

00:01:32.150 --> 00:01:33.700
children were born in '93?

00:01:33.700 --> 00:01:36.240
Well they had 1 child in '92 and
they had 1 child in '93,

00:01:36.240 --> 00:01:39.870
so no children were
born in '93.

00:01:39.870 --> 00:01:40.750
I think you see what
I'm doing.

00:01:40.750 --> 00:01:43.210
I'm getting the difference
between one year and the next

00:01:43.210 --> 00:01:44.240
because that's how
many children are

00:01:44.240 --> 00:01:46.420
born in that year.

00:01:46.420 --> 00:01:50.510
From '93 to '94, we went from
1 to 2 children, so 1 child

00:01:50.510 --> 00:01:51.470
was born that year.

00:01:51.470 --> 00:01:53.170
So there was 1 child born.

00:01:53.170 --> 00:01:54.630
Same thing in '95.

00:01:54.630 --> 00:01:56.830
The number of children
increased by 1.

00:01:56.830 --> 00:01:58.710
So there was 1 child born.

00:01:58.710 --> 00:02:02.180
'96, the number of children
stayed the same.

00:02:02.180 --> 00:02:03.260
No new children were born.

00:02:03.260 --> 00:02:05.220
We had 3 at the end of
'95, and we had 3

00:02:05.220 --> 00:02:06.100
at the end of '96.

00:02:06.100 --> 00:02:07.500
So no new children were born.

00:02:07.500 --> 00:02:09.199
So that's a 0.

00:02:09.199 --> 00:02:11.986
And then in '97, we go from
having 3 children at the end

00:02:11.986 --> 00:02:14.300
of '96 to having 5.

00:02:14.300 --> 00:02:17.640
So 2 children must have
been born that year.

00:02:17.640 --> 00:02:20.620
And then in '98, we had the
same number of children.

00:02:20.620 --> 00:02:22.950
And of course, they should have
said in the question, but

00:02:22.950 --> 00:02:25.780
we can assume that no children
died during the

00:02:25.780 --> 00:02:27.650
course of this problem.

00:02:27.650 --> 00:02:28.320
I don't want to be morbid.

00:02:28.320 --> 00:02:32.890
But anyway, so no children
were born in '98, so 0.

00:02:32.890 --> 00:02:35.850
So the question is, what year
were the twins born?

00:02:35.850 --> 00:02:39.930
Well twins, that's 2 children,
if my biology is correct.

00:02:39.930 --> 00:02:45.500
So the year is 1997, which
is choice number D.

00:02:45.500 --> 00:02:48.600
I feel very bad having to erase
this beautiful drawing,

00:02:48.600 --> 00:02:50.695
but the things I do in the
name of education.

00:02:56.780 --> 00:02:59.780
Number 5-- no drawing
for Sal in number 5.

00:02:59.780 --> 00:03:03.860
The average arithmetic
mean of X and Y is 5.

00:03:03.860 --> 00:03:08.940
So average-- let me write mean
because if has fewer letters.

00:03:08.940 --> 00:03:11.610
They should have another math
term called nice I think.

00:03:11.610 --> 00:03:17.270
But the mean of x and
y is equal to 5.

00:03:17.270 --> 00:03:18.100
And what does that mean?

00:03:18.100 --> 00:03:20.540
That means that the average-- so
we also know that x plus y

00:03:20.540 --> 00:03:24.750
over 2 is equal to 5-- I just
took the average of them.

00:03:24.750 --> 00:03:28.100
And the average of
x, y and z is 8.

00:03:28.100 --> 00:03:31.870
What is the value of z?

00:03:31.870 --> 00:03:32.950
Fascinating.

00:03:32.950 --> 00:03:42.790
So we also know that the mean
of x, y and z is equal to 8.

00:03:42.790 --> 00:03:45.930
And then they want to
know what is z?

00:03:45.930 --> 00:03:48.750
Well the mean of x, y and z is
equal to 8-- that's just if we

00:03:48.750 --> 00:03:49.580
took the average of them.

00:03:49.580 --> 00:03:59.190
That's x plus y plus z
over 3 is equal to 8.

00:03:59.190 --> 00:04:00.500
This is the exact same
way of saying this.

00:04:00.500 --> 00:04:04.470
You can review the average
play-- I think it's in my

00:04:04.470 --> 00:04:06.630
pre-algebra playlist
if you don't

00:04:06.630 --> 00:04:08.210
remember how to take averages.

00:04:08.210 --> 00:04:11.570
But how do we solve for z here,
because we have three

00:04:11.570 --> 00:04:13.850
variables and only two
equations, it seems very

00:04:13.850 --> 00:04:14.850
complicated.

00:04:14.850 --> 00:04:16.600
But I'll show you a trick,
and this is actually a

00:04:16.600 --> 00:04:18.750
very common SAT trick.

00:04:18.750 --> 00:04:23.360
So you take this first equation
and we can't solve

00:04:23.360 --> 00:04:25.380
for x and we can't solve
for y, but we can

00:04:25.380 --> 00:04:27.790
solve for x plus y.

00:04:27.790 --> 00:04:31.590
So we just multiply both sides
of that equation by 2 and we

00:04:31.590 --> 00:04:36.350
get x plus y is equal to 10.

00:04:36.350 --> 00:04:41.640
x plus y is equal to 10.

00:04:41.640 --> 00:04:47.960
And then we can substitute
the whole x plus y here.

00:04:47.960 --> 00:04:50.050
This is something you seldom
see in your algebra classes

00:04:50.050 --> 00:04:52.540
because you're always solving
for one variable for other.

00:04:52.540 --> 00:04:56.170
Now you can almost use x
plus y as a variable.

00:04:56.170 --> 00:04:57.610
Like the whole thing
is a variable.

00:04:57.610 --> 00:04:59.200
It's kind of strange,
but that's

00:04:59.200 --> 00:05:01.550
how you do this problem.

00:05:01.550 --> 00:05:03.050
Algebra, there's nothing
fancy here.

00:05:03.050 --> 00:05:06.510
I'm saying x plus y is equal to
10, literally in English.

00:05:06.510 --> 00:05:09.600
So we could say well this
x plus y is equal to 10.

00:05:09.600 --> 00:05:17.540
So we could re-write this top
equation as 10 plus z over 3

00:05:17.540 --> 00:05:20.460
is equal to 8.

00:05:20.460 --> 00:05:25.250
And then if we multiply both
sides by 3 you get 10 plus z

00:05:25.250 --> 00:05:27.750
is equal to 24.

00:05:27.750 --> 00:05:30.600
And then you subtract 10 from
both sides and you get z is

00:05:30.600 --> 00:05:33.170
equal to 14, and we are done.

00:05:33.170 --> 00:05:34.620
That wasn't too bad, was it?

00:05:34.620 --> 00:05:38.130
Good, the choice exists,
choice B for number 5.

00:05:38.130 --> 00:05:39.050
All right, number 6.

00:05:39.050 --> 00:05:40.540
So I'm drawing for Sal.

00:05:48.820 --> 00:05:51.490
I like these, these are fun.

00:05:51.490 --> 00:05:52.740
Let me draw in magenta.

00:06:02.190 --> 00:06:03.930
That's the big figure.

00:06:03.930 --> 00:06:05.900
In the figure above-- I don't
know how to change my

00:06:05.900 --> 00:06:07.150
background color.

00:06:09.740 --> 00:06:11.500
In the figure above,
a small square is

00:06:11.500 --> 00:06:12.990
inside a larger square.

00:06:12.990 --> 00:06:16.080
What is the area in terms of
x of the shaded region?

00:06:16.080 --> 00:06:24.390
Well let me just work with what
I have. So they have a

00:06:24.390 --> 00:06:26.040
smaller one here,
then a smaller.

00:06:28.720 --> 00:06:34.040
And then let me use
this thing.

00:06:34.040 --> 00:06:35.270
Create a shaded region.

00:06:35.270 --> 00:06:36.400
Let me create a shaded region.

00:06:36.400 --> 00:06:38.665
Look at that-- look at the
amazing tool I'm using.

00:06:38.665 --> 00:06:40.005
It's called paint.

00:06:43.180 --> 00:06:44.580
So what was I doing.

00:06:44.580 --> 00:06:46.380
Oh, this is x.

00:06:46.380 --> 00:06:48.040
Oh, I'm using the same color.

00:06:48.040 --> 00:06:50.480
Let me use a vivid color.

00:06:50.480 --> 00:06:54.300
One side of this is x, one
side of this is x.

00:06:54.300 --> 00:07:00.420
And then the sides of the
big square are 5 and 5.

00:07:00.420 --> 00:07:03.490
And they just want to know, what
is the area in terms of x

00:07:03.490 --> 00:07:04.990
of the shaded region?

00:07:04.990 --> 00:07:08.700
Well the shaded region, that's
this brown area, the area of

00:07:08.700 --> 00:07:12.490
that is going to be the area
of this entire square minus

00:07:12.490 --> 00:07:14.590
the area of the smaller square,
right, because we took

00:07:14.590 --> 00:07:17.390
this out of the shaded region.

00:07:17.390 --> 00:07:21.180
So the area of the big square--
well the area of a

00:07:21.180 --> 00:07:25.500
square or a rectangle is just
the height times the width.

00:07:25.500 --> 00:07:29.570
So that's 5 times 5, so it's
25, minus the area of the

00:07:29.570 --> 00:07:30.880
smaller square.

00:07:30.880 --> 00:07:35.140
And that's just going to be x
times x, which is x squared.

00:07:35.140 --> 00:07:38.580
That equals the area, and
that is choice E.

00:07:38.580 --> 00:07:42.070
All that drawing for such
an easy problem.

00:07:42.070 --> 00:07:44.860
I hope you don't find that I'm
being cocky when I call it an

00:07:44.860 --> 00:07:45.750
easy problem.

00:07:45.750 --> 00:07:47.820
If you find it difficult,
it wasn't that easy, I

00:07:47.820 --> 00:07:48.440
understand.

00:07:48.440 --> 00:07:50.650
Anyway, number 7.

00:07:50.650 --> 00:07:52.810
Let me erase my drawing.

00:07:56.840 --> 00:07:59.130
If rstv equals 1.

00:07:59.130 --> 00:08:00.380
Let me write that down.

00:08:05.830 --> 00:08:07.455
And stuv equals 0.

00:08:17.280 --> 00:08:18.970
Which of the following
must be true?

00:08:18.970 --> 00:08:19.980
Let me write these down.

00:08:19.980 --> 00:08:23.130
So they're saying choice
A-- I'm going to

00:08:23.130 --> 00:08:24.150
write all the choices.

00:08:24.150 --> 00:08:31.650
Choice A, r is less than 1.

00:08:31.650 --> 00:08:36.960
Choice B is s is less than 1.

00:08:36.960 --> 00:08:44.260
Choice C is t is greater
than 1/2.

00:08:44.260 --> 00:08:50.790
Choice D, u is equal to 0.

00:08:50.790 --> 00:08:57.480
And choice E, v is equal to 0.

00:08:57.480 --> 00:09:00.500
So which of the following
must be true?

00:09:00.500 --> 00:09:04.890
Well the first thing we know,
we know rstv is equal to 1.

00:09:04.890 --> 00:09:09.270
So we know that none of rst or
v can be 0, because if any of

00:09:09.270 --> 00:09:12.180
them were 0 than when you
multiply them out, they

00:09:12.180 --> 00:09:13.190
couldn't equal 0, right?

00:09:13.190 --> 00:09:19.310
So rstv all must be
some non-0 number.

00:09:19.310 --> 00:09:23.000
So if we go back here, we know
that s is non-- and then if we

00:09:23.000 --> 00:09:26.940
go into this equation we know
that one of them must be 0, at

00:09:26.940 --> 00:09:29.880
least 1, because when you
multiply them you get 0.

00:09:29.880 --> 00:09:32.430
Because 0 times any
number is 0.

00:09:32.430 --> 00:09:38.630
So we know that s is not
0 because it's up here.

00:09:38.630 --> 00:09:40.990
Because we said all of these
numbers cannot be 0.

00:09:40.990 --> 00:09:43.250
We know t cannot be 0 because
it's up here.

00:09:43.250 --> 00:09:45.230
Because if it was 0, this
would have been 0.

00:09:45.230 --> 00:09:48.800
And we know v cannot be 0
because it's up here.

00:09:48.800 --> 00:09:53.080
And all we're left with is
that u is equal to 0.

00:09:53.080 --> 00:09:55.460
I'll see you in the
next video.


SAT prep: test 1 section3 part 1

WEBVTT
Kind: captions
Language: en

00:00:00.830 --> 00:00:01.610
Good morning.

00:00:01.610 --> 00:00:03.630
Well today is what I hope is an

00:00:03.630 --> 00:00:04.740
exciting day at Khan Academy.

00:00:04.740 --> 00:00:07.860
We're going to start a new set
of, I guess we could call

00:00:07.860 --> 00:00:11.000
them, lectures.

00:00:11.000 --> 00:00:12.780
But what we're going to do is
I'm actually going to work

00:00:12.780 --> 00:00:15.670
through just a bunch of SAT
problems. So what this is

00:00:15.670 --> 00:00:19.850
going to be is like SAT prep for
free, or Princeton Review

00:00:19.850 --> 00:00:23.380
or Kaplan for free, and I hope
neither of them sue me for

00:00:23.380 --> 00:00:24.170
using their names.

00:00:24.170 --> 00:00:27.320
But anyway, what I want you to
do is, if you haven't already,

00:00:27.320 --> 00:00:31.330
I want you to buy this
book right here.

00:00:31.330 --> 00:00:33.330
The Official SAT Study Guide.

00:00:33.330 --> 00:00:35.526
Let me make sure it all fits
into the-- you'll see it in

00:00:35.526 --> 00:00:36.980
the bookstore.

00:00:36.980 --> 00:00:39.210
I hope I'm not doing any kind
of copyright infringement.

00:00:39.210 --> 00:00:39.850
I'm not a lawyer.

00:00:39.850 --> 00:00:41.050
I don't know all the rules.

00:00:41.050 --> 00:00:43.610
But I'm pretty sure as long as
I give them credit and that

00:00:43.610 --> 00:00:45.460
you know that all the problems
we're going to be doing are

00:00:45.460 --> 00:00:47.490
from their book and I'm just
going to work through them,

00:00:47.490 --> 00:00:51.770
hopefully that shouldn't be
something that's wrong.

00:00:51.770 --> 00:00:54.800
Every day-- well not every
day, but as often as

00:00:54.800 --> 00:00:59.730
possible-- I'm going to work
through all of the problems in

00:00:59.730 --> 00:01:01.130
each of the math sections.

00:01:01.130 --> 00:01:05.530
So the best thing for you to do
is for the same section, do

00:01:05.530 --> 00:01:07.560
them ahead of time for yourself,
try to do as many as

00:01:07.560 --> 00:01:11.090
you can, score your test, and
then whichever problems you

00:01:11.090 --> 00:01:14.550
don't know, you can just watch
on Khan Academy and hopefully

00:01:14.550 --> 00:01:16.810
understand them or re-understand
them.

00:01:16.810 --> 00:01:18.170
So with that said, let's
get started.

00:01:18.170 --> 00:01:19.930
So I'm using this
book right here.

00:01:19.930 --> 00:01:26.830
Official SAT Study Guide, and
I am starting on page 395.

00:01:26.830 --> 00:01:29.850
This is section three,
I believe, of

00:01:29.850 --> 00:01:32.550
the first math section.

00:01:32.550 --> 00:01:35.240
This is page 395.

00:01:35.240 --> 00:01:36.330
So sometimes I'll
write the whole

00:01:36.330 --> 00:01:38.160
problem, sometimes I won't.

00:01:38.160 --> 00:01:41.000
I'm going to assume that
you have the textbook.

00:01:41.000 --> 00:01:44.920
So with that said let's begin.

00:01:44.920 --> 00:01:55.150
So the first problem, problem
number one, it says, if 2x

00:01:55.150 --> 00:02:07.610
plus 3 is equal to 9, then
what is 4x minus 3.

00:02:10.750 --> 00:02:14.130
Well this is just a very
basic algebra problem.

00:02:14.130 --> 00:02:16.210
If you don't remember how to
do this you should probably

00:02:16.210 --> 00:02:19.640
review the level one and level
two linear equations lectures

00:02:19.640 --> 00:02:22.110
that I have in the
algebra playlist.

00:02:22.110 --> 00:02:25.180
But the way we do this, we could
solve for x here, so we

00:02:25.180 --> 00:02:27.540
can just subtract 3
from both sides.

00:02:27.540 --> 00:02:34.920
2x plus 3 minus 3, so you get
a 0 here is equal to 6.

00:02:34.920 --> 00:02:37.900
And x is equal to 3.

00:02:37.900 --> 00:02:45.610
So if you do 4x minus 3, that's
4 times 3 minus 3.

00:02:45.610 --> 00:02:48.790
Right, that's 12 minus 3.

00:02:48.790 --> 00:02:51.660
And that also equals 9.

00:02:51.660 --> 00:02:54.050
And there are some, you could
have kind of done this a

00:02:54.050 --> 00:02:54.840
little bit faster.

00:02:54.840 --> 00:02:57.840
But this is the one way that you
would make sure that you

00:02:57.840 --> 00:02:59.190
got the right answer.

00:02:59.190 --> 00:03:02.840
And on this test it's B, 9.

00:03:02.840 --> 00:03:04.090
Number two.

00:03:07.900 --> 00:03:10.660
There are 8 sections of seats
in an auditorium.

00:03:10.660 --> 00:03:12.670
So I will write-- let me do it
in another color just keep

00:03:12.670 --> 00:03:16.200
things interesting--
8 sections.

00:03:20.260 --> 00:03:23.170
There are 8 sections of seats
in an auditorium.

00:03:23.170 --> 00:03:27.200
Each section contains
150 seats, but no

00:03:27.200 --> 00:03:35.100
more than 250 seats.

00:03:35.100 --> 00:03:36.970
So if we say S is seats.

00:03:36.970 --> 00:03:39.410
Let me say S is seats.

00:03:39.410 --> 00:03:46.280
Seats are greater than or equal
to 150 and less than or

00:03:46.280 --> 00:03:47.760
equal to 200.

00:03:47.760 --> 00:03:49.822
This is seats per section.

00:03:49.822 --> 00:03:52.900
I hope I'm not confusing you.

00:03:52.900 --> 00:03:57.290
Which of the following could be
the number of seats in this

00:03:57.290 --> 00:03:58.370
auditorium.

00:03:58.370 --> 00:04:01.910
So before I even look at the
choices, I mean you could put

00:04:01.910 --> 00:04:04.020
in the choices, you just have to
figure out, well what's the

00:04:04.020 --> 00:04:08.060
minimum number of seats
in this auditorium?

00:04:08.060 --> 00:04:10.340
Well the minimum number
is if every section

00:04:10.340 --> 00:04:13.420
only has 150 seats.

00:04:13.420 --> 00:04:17.760
So the minimum number is 8 times
150, 8 sections each of

00:04:17.760 --> 00:04:19.209
them having 150 seats.

00:04:19.209 --> 00:04:22.490
Because this is the case in
which every section has the

00:04:22.490 --> 00:04:23.480
smallest number of seats.

00:04:23.480 --> 00:04:32.940
So 150 times 8-- 0, 40.

00:04:32.940 --> 00:04:37.280
1 times 8 is 8 plus 4 is 12.

00:04:37.280 --> 00:04:41.930
So this is the minimum number
of seats in the auditorium.

00:04:41.930 --> 00:04:44.260
And then the maximum number
would be if we assume every

00:04:44.260 --> 00:04:46.490
section had the maximum
number of seats.

00:04:46.490 --> 00:04:51.190
So that would be 200 times 8.

00:04:51.190 --> 00:04:51.700
That's easy.

00:04:51.700 --> 00:04:55.400
2 times 8 is 1,600.

00:04:55.400 --> 00:05:01.130
So we know that seats is going
to be greater than or equal to

00:05:01.130 --> 00:05:02.970
this and less than
or equal to this.

00:05:02.970 --> 00:05:05.910
And what choice falls
under that range?

00:05:05.910 --> 00:05:07.870
Choice A doesn't, it's 800.

00:05:07.870 --> 00:05:11.050
Choice B doesn't, choice
C doesn't.

00:05:11.050 --> 00:05:14.140
Choice D, 1,300, does fit in
that range, and we know that

00:05:14.140 --> 00:05:15.570
choice E doesn't fit
in that range.

00:05:15.570 --> 00:05:16.520
So we're done.

00:05:16.520 --> 00:05:23.730
It's D, 1,300 seats
is a possibility.

00:05:23.730 --> 00:05:26.610
And we turn the page and
we a graphical problem.

00:05:26.610 --> 00:05:28.020
Let's see what we could
do with this.

00:05:33.850 --> 00:05:36.920
Note: Figure not drawn to scale
in the figure above.

00:05:36.920 --> 00:05:38.060
Let me draw this.

00:05:38.060 --> 00:05:42.900
I think I'll have to explain
how it's done.

00:05:42.900 --> 00:05:46.640
I will use sky blue.

00:05:46.640 --> 00:05:48.300
So the figures-- let
me draw that line.

00:05:52.130 --> 00:05:56.310
No, let me do it better
than that.

00:05:56.310 --> 00:05:58.850
Draw a thicker line than
that so you can see it.

00:06:01.420 --> 00:06:05.110
So they have a line like that.

00:06:05.110 --> 00:06:06.860
And they have another
line that goes up

00:06:06.860 --> 00:06:08.110
and down like that.

00:06:12.920 --> 00:06:14.470
And then they had a bunch
of points, so let

00:06:14.470 --> 00:06:15.720
me draw those points.

00:06:19.280 --> 00:06:21.900
I will draw them in
a different color.

00:06:21.900 --> 00:06:47.160
So we have point A, point B,
point C, point D, point E.

00:06:47.160 --> 00:06:48.920
And they call this horizontal
line, they're

00:06:48.920 --> 00:06:52.080
calling this line L.

00:06:52.080 --> 00:06:53.740
And then no, they have another
point up here.

00:06:53.740 --> 00:06:56.570
Point X.

00:06:56.570 --> 00:06:59.740
In the figure above XC is
perpendicular to L.

00:06:59.740 --> 00:07:02.740
So this XC is perpendicular
to line L.

00:07:02.740 --> 00:07:05.100
So we know that this is
a 90 degree angle.

00:07:05.100 --> 00:07:05.900
And I think they draw it.

00:07:05.900 --> 00:07:07.610
Yes, they draw that.

00:07:07.610 --> 00:07:09.950
Which of the following
line segments has

00:07:09.950 --> 00:07:10.690
the greatest length?

00:07:10.690 --> 00:07:12.870
Oh, and in the figure they also
tell you the measurements

00:07:12.870 --> 00:07:14.260
of the different lengths.

00:07:14.260 --> 00:07:17.890
So they say from A to B is 3.

00:07:17.890 --> 00:07:20.980
From B to C is 2.

00:07:20.980 --> 00:07:24.020
C to D is 3.

00:07:24.020 --> 00:07:26.970
D to E is 1.

00:07:26.970 --> 00:07:31.930
And from C to X is 8.

00:07:31.930 --> 00:07:34.760
Which of the following line
segments that are not shown

00:07:34.760 --> 00:07:36.620
has the greatest length?

00:07:36.620 --> 00:07:38.710
So the first choice--
I'll write the

00:07:38.710 --> 00:07:40.220
choices down actually.

00:07:40.220 --> 00:07:43.060
Choice A is line segment XA.

00:07:45.620 --> 00:07:50.780
Choice B is line segment XB.

00:07:50.780 --> 00:07:52.090
Well this is following
a pattern.

00:07:52.090 --> 00:07:55.210
Choice C is line segment XC.

00:07:55.210 --> 00:08:01.110
Choice D is line segment XD,
and then choice E is line

00:08:01.110 --> 00:08:02.360
segment XE.

00:08:04.810 --> 00:08:08.490
So which of the following
segments has

00:08:08.490 --> 00:08:11.870
the greatest length?

00:08:11.870 --> 00:08:18.370
So just eye-balling it, you'll
know that XB is definitely

00:08:18.370 --> 00:08:21.050
going to be shorter.

00:08:21.050 --> 00:08:29.090
XB is definitely going to
be shorter than XA.

00:08:29.090 --> 00:08:29.950
Why is that?

00:08:29.950 --> 00:08:31.600
Because if you used--
well one, you could

00:08:31.600 --> 00:08:32.390
just eye-ball it.

00:08:32.390 --> 00:08:33.360
I know it's not drawn
to scale.

00:08:33.360 --> 00:08:36.809
But you definitely know that
this length, this base, is

00:08:36.809 --> 00:08:39.409
longer than this base
right here.

00:08:39.409 --> 00:08:42.200
So we know it's not
going to be XB.

00:08:42.200 --> 00:08:45.170
We know XB is shorter
than definitely XA.

00:08:45.170 --> 00:08:47.750
So we can cross that
one out, right?

00:08:47.750 --> 00:08:50.330
Now what we don't know-- and we
could do the same thing on

00:08:50.330 --> 00:08:54.620
the other side of this
perpendicular line.

00:08:54.620 --> 00:09:01.100
We know that XD is shorter
than XE, right?

00:09:01.100 --> 00:09:03.920
Same reason, because we know
this base is longer.

00:09:03.920 --> 00:09:05.880
If you use a Pythagorean
theorem, you would know that

00:09:05.880 --> 00:09:09.000
this hypotenuse is going to
be longer than this one.

00:09:09.000 --> 00:09:11.190
And you can review Pythagorean
theorem and get familiar with

00:09:11.190 --> 00:09:13.190
the triangles if that doesn't
make a lot of sense.

00:09:13.190 --> 00:09:17.790
But that tells us that it's
definitely not going to be XD.

00:09:17.790 --> 00:09:19.260
And we also know it's
not going to be XC.

00:09:19.260 --> 00:09:21.970
This is the shortest
of all of them.

00:09:21.970 --> 00:09:23.740
And you could eye-ball that.

00:09:23.740 --> 00:09:30.160
So it's either going
to be XE or XA.

00:09:30.160 --> 00:09:32.880
And the way we figure out
which one has the longer

00:09:32.880 --> 00:09:36.620
hypotenuse is we say, well they
all have the same height,

00:09:36.620 --> 00:09:38.720
so which one has the
longer base?

00:09:38.720 --> 00:09:42.730
So this one has a base, this
triangle right here-- if we

00:09:42.730 --> 00:09:47.910
say this triangle-- has a base
of 4, and this triangle here

00:09:47.910 --> 00:09:49.880
has a base of 5.

00:09:49.880 --> 00:09:54.990
So this hypotenuse will be the
longest. XA is the answer.

00:09:54.990 --> 00:09:57.510
I will see you in
the next video.


Introduction to logarithm properties (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.630
PROFESSOR: Welcome back.

00:00:01.630 --> 00:00:04.060
I'm going to show you the last
two logarithm properties now.

00:00:04.060 --> 00:00:07.500
So this one-- and I always
found this one to be in some

00:00:07.500 --> 00:00:08.460
ways the most obvious one.

00:00:08.460 --> 00:00:11.380
But don't feel bad if
it's not obvious.

00:00:11.380 --> 00:00:12.690
Maybe it'll take a little
bit of introspection.

00:00:12.690 --> 00:00:14.770
And I encourage you to really
experiment with all these

00:00:14.770 --> 00:00:16.760
logarithm properties, because
that's the only way that

00:00:16.760 --> 00:00:17.970
you'll really learn them.

00:00:17.970 --> 00:00:21.510
And the point of math isn't
just to pass the next exam, or

00:00:21.510 --> 00:00:22.560
to get an A on the next exam.

00:00:22.560 --> 00:00:24.990
The point of math is to
understand math so you can

00:00:24.990 --> 00:00:28.360
actually apply it in life later
on and not have to relearn

00:00:28.360 --> 00:00:29.700
everything every time.

00:00:29.700 --> 00:00:36.040
So the next logarithm property
is, if I have A times the

00:00:36.040 --> 00:00:43.060
logarithm base B of C, if I
have A times this whole thing,

00:00:43.060 --> 00:00:59.170
that that equals logarithm
base B of C to the A power.

00:00:59.170 --> 00:01:00.530
Fascinating.

00:01:00.530 --> 00:01:02.300
So let's see if this works out.

00:01:02.300 --> 00:01:17.220
So let's say if I have 3
times logarithm base 2 of 8.

00:01:17.220 --> 00:01:19.990
So this property tells us that
this is going to be the same

00:01:19.990 --> 00:01:30.250
thing as logarithm base 2
of 8 to the third power.

00:01:30.250 --> 00:01:32.200
And that's the same thing.

00:01:32.200 --> 00:01:38.150
Well, that's the same thing
as-- we could figure it out.

00:01:38.150 --> 00:01:39.500
So let's see what this is.

00:01:39.500 --> 00:01:43.480
3 times log base--
what's log base 2 of 8?

00:01:43.480 --> 00:01:45.540
The reason why I kind of
hesitated a second ago is

00:01:45.540 --> 00:01:48.760
because every time I want to
figure something out, I

00:01:48.760 --> 00:01:52.780
implicitly want to use log and
exponential rules to do it.

00:01:52.780 --> 00:01:54.200
So I'm trying to avoid that.

00:01:54.200 --> 00:01:55.670
Anyway, going back.

00:01:55.670 --> 00:01:57.360
What is this?

00:01:57.360 --> 00:01:59.110
2 to the what power is 8?

00:01:59.110 --> 00:02:00.570
2 to the third
power is 8, right?

00:02:00.570 --> 00:02:01.890
So that's 3.

00:02:01.890 --> 00:02:05.060
We have this 3 here,
so 3 times 3.

00:02:05.060 --> 00:02:09.310
So this thing right
here should equal 9.

00:02:09.310 --> 00:02:11.820
If this equals 9, then we know
that this property works

00:02:11.820 --> 00:02:12.970
at least for this example.

00:02:12.970 --> 00:02:14.980
You don't know if it works for
all examples, and for that

00:02:14.980 --> 00:02:17.990
maybe you'd want to look at
the proof we have in

00:02:17.990 --> 00:02:18.630
the other videos.

00:02:18.630 --> 00:02:20.970
But that's kind of a
more advanced topic.

00:02:20.970 --> 00:02:22.890
But the important thing
first is just to

00:02:22.890 --> 00:02:24.520
understand how to use it.

00:02:24.520 --> 00:02:27.770
Let's see, what is 2
to the ninth power?

00:02:27.770 --> 00:02:29.250
Well it's going to be
some large number.

00:02:29.250 --> 00:02:32.740
Actually, I know what
it is-- it's 256.

00:02:32.740 --> 00:02:34.900
Because in the last video we
figured out that 2 to the

00:02:34.900 --> 00:02:38.110
eighth was equal to 256.

00:02:38.110 --> 00:02:43.180
So 2 to the ninth
should be 512.

00:02:43.180 --> 00:02:44.890
So 2 to the ninth
should be 512.

00:02:44.890 --> 00:02:50.820
So if 8 to the third is also
512 then we are correct, right?

00:02:50.820 --> 00:02:57.780
Because log base 2 of 512
is going to be equal to 9.

00:02:57.780 --> 00:02:58.690
What's 8 to the third?

00:02:58.690 --> 00:03:00.660
It's 64-- right.

00:03:00.660 --> 00:03:05.440
8 squared is 64, so 8
cubed-- let's see.

00:03:05.440 --> 00:03:07.740
4 times 2 is 3.

00:03:07.740 --> 00:03:10.420
6 times 8-- looks
like it's 512.

00:03:10.420 --> 00:03:12.150
Correct.

00:03:12.150 --> 00:03:13.660
And there's other ways
you could have done it.

00:03:13.660 --> 00:03:16.030
Because you could have said
8 to the third is the same

00:03:16.030 --> 00:03:16.950
thing as 2 to the ninth.

00:03:16.950 --> 00:03:18.140
How do we know that?

00:03:18.140 --> 00:03:24.440
Well, 8 to the third is
equal to 2 to the third

00:03:24.440 --> 00:03:25.240
to the third, right?

00:03:25.240 --> 00:03:28.030
I just rewrote 8.

00:03:28.030 --> 00:03:30.510
And we know from our exponent
rules that 2 to the third

00:03:30.510 --> 00:03:35.290
to the third is the same
thing as 2 to the ninth.

00:03:35.290 --> 00:03:38.420
And actually it's this exponent
property, where you can

00:03:38.420 --> 00:03:40.130
multiply-- when you take
something to exponent and then

00:03:40.130 --> 00:03:42.460
take that to an exponent, and
you can essentially just

00:03:42.460 --> 00:03:45.460
multiply the exponents-- that's
the exponent property that

00:03:45.460 --> 00:03:47.850
actually leads to this
logarithm property.

00:03:50.540 --> 00:03:52.860
But I'm not going to dwell
on that too much in

00:03:52.860 --> 00:03:54.080
this presentation.

00:03:54.080 --> 00:03:56.430
There's a whole video on
kind of proving it a

00:03:56.430 --> 00:03:58.480
little bit more formally.

00:03:58.480 --> 00:04:01.830
The next logarithm property I'm
going to show you-- and then

00:04:01.830 --> 00:04:04.580
I'll review everything and
maybe do some examples.

00:04:04.580 --> 00:04:09.020
This is probably the single
most useful logarithm property

00:04:09.020 --> 00:04:11.570
if you are a calculator addict.

00:04:11.570 --> 00:04:13.700
And I'll show you why.

00:04:13.700 --> 00:04:31.950
So let's say I have log base B
of A is equal to log base C of

00:04:31.950 --> 00:04:40.480
A divided by log base C of B.

00:04:40.480 --> 00:04:43.880
Now why is this a useful
property if you are

00:04:43.880 --> 00:04:45.030
calculator addict?

00:04:45.030 --> 00:04:47.520
Well, let's say you go
class, and there's a quiz.

00:04:47.520 --> 00:04:51.660
The teacher says, you can use
your calculator, and using your

00:04:51.660 --> 00:05:03.390
calculator I want you to figure
out the log base 17 of 357.

00:05:03.390 --> 00:05:07.870
And you will scramble and look
for the log base 17 button on

00:05:07.870 --> 00:05:09.530
your calculator,
and not find it.

00:05:09.530 --> 00:05:14.150
Because there is no log base
17 button on your calculator.

00:05:14.150 --> 00:05:17.410
You'll probably either have
a log button or you'll

00:05:17.410 --> 00:05:19.050
have an ln button.

00:05:19.050 --> 00:05:21.800
And just so you know, the log
button on your calculator

00:05:21.800 --> 00:05:24.520
is probably base 10.

00:05:24.520 --> 00:05:28.470
And your ln button on
your calculator is

00:05:28.470 --> 00:05:29.560
going to be base e.

00:05:29.560 --> 00:05:31.280
For those you who aren't
familiar with e, don't worry

00:05:31.280 --> 00:05:34.080
about it, but it's 2.71
something something.

00:05:34.080 --> 00:05:34.920
It's a number.

00:05:34.920 --> 00:05:38.000
It's nothing-- it's an amazing
number, but we'll talk

00:05:38.000 --> 00:05:41.075
more about that in a
future presentation.

00:05:41.075 --> 00:05:44.580
But so there's only two bases
you have on your calculator.

00:05:44.580 --> 00:05:48.370
So if you want to figure out
another base logarithm,

00:05:48.370 --> 00:05:50.020
you use this property.

00:05:50.020 --> 00:05:54.170
So if you're given this on an
exam, you can very confidently

00:05:54.170 --> 00:05:57.940
say, oh, well that is just the
same thing as-- you'd have to

00:05:57.940 --> 00:06:01.890
switch to your yellow color in
order to act with confidence--

00:06:01.890 --> 00:06:05.660
log base-- we could
do either e or 10.

00:06:05.660 --> 00:06:11.290
We could say that's the same
thing as log base 10 of 357

00:06:11.290 --> 00:06:15.810
divided by log base 10 of 17.

00:06:15.810 --> 00:06:19.740
So you literally could just
type in 357 in your calculator

00:06:19.740 --> 00:06:21.060
and press the log button and
you're going to get

00:06:21.060 --> 00:06:22.230
bada bada bam.

00:06:22.230 --> 00:06:24.440
Then you can clear it, or if
you know how to use the

00:06:24.440 --> 00:06:25.540
parentheses on your calculator,
you could do that.

00:06:25.540 --> 00:06:28.610
But then you type 17 on your
calculator, press the log

00:06:28.610 --> 00:06:29.530
button, go to bada bada bam.

00:06:29.530 --> 00:06:31.400
And then you just divide them,
and you get your answer.

00:06:31.400 --> 00:06:35.050
So this is a super
useful property for

00:06:35.050 --> 00:06:37.370
calculator addicts.

00:06:37.370 --> 00:06:40.930
And once again, I'm not going
to go into a lot of depth.

00:06:40.930 --> 00:06:46.280
This one, to me it's the most
useful, but it doesn't

00:06:46.280 --> 00:06:49.375
completely-- it does fall
out of, obviously, the

00:06:49.375 --> 00:06:50.050
exponent properties.

00:06:50.050 --> 00:06:53.940
But it's hard for me to
describe the intuition simply,

00:06:53.940 --> 00:06:55.930
so you probably want to watch
the proof on it, if you don't

00:06:55.930 --> 00:06:58.330
believe why this happens.

00:06:58.330 --> 00:07:00.740
But anyway, with all of those
aside, and this is probably the

00:07:00.740 --> 00:07:03.130
one you're going to be using
the most in everyday life.

00:07:03.130 --> 00:07:05.850
I still use this in my job.

00:07:05.850 --> 00:07:09.400
Just so you know
logarithms are useful.

00:07:09.400 --> 00:07:14.110
Let's do some examples.

00:07:14.110 --> 00:07:16.360
Let's just let's just
rewrite a bunch of

00:07:16.360 --> 00:07:19.240
things in simpler forms.

00:07:19.240 --> 00:07:33.840
So if I wanted to rewrite the
log base 2 of the square root

00:07:33.840 --> 00:07:37.860
of-- let me think of something.

00:07:37.860 --> 00:07:50.560
Of 32 divided by the cube-- no,
I'll just take the square root.

00:07:50.560 --> 00:07:54.120
Divided by the
square root of 8.

00:07:54.120 --> 00:07:59.090
How can I rewrite this so
it's reasonably not messy?

00:07:59.090 --> 00:08:00.210
Well let's think about this.

00:08:00.210 --> 00:08:04.460
This is the same thing, this
is equal to-- I don't know

00:08:04.460 --> 00:08:05.800
if I'll move vertically
or horizontally.

00:08:05.800 --> 00:08:07.320
I'll move vertically.

00:08:07.320 --> 00:08:13.430
This is the same thing as
the log base 2 of 32 over

00:08:13.430 --> 00:08:17.750
the square root of 8 to
the 1/2 power, right?

00:08:17.750 --> 00:08:20.350
And we know from our logarithm
properties, the third one we

00:08:20.350 --> 00:08:26.560
learned, that that is the same
thing as 1/2 times the

00:08:26.560 --> 00:08:33.840
logarithm of 32 divided by the
square root of 8, right?

00:08:33.840 --> 00:08:35.830
I just took the exponent and
made that the coefficient

00:08:35.830 --> 00:08:36.585
on the entire thing.

00:08:36.585 --> 00:08:39.410
And we learned that in the
beginning of this video.

00:08:39.410 --> 00:08:41.650
And now we have a little
quotient here, right?

00:08:41.650 --> 00:08:45.340
Logarithm of 32 divided by
logarithm of square root of 8.

00:08:45.340 --> 00:08:47.030
Well, we can use our
other logarithm-- let's

00:08:47.030 --> 00:08:49.480
keep the 1/2 out.

00:08:49.480 --> 00:08:56.470
That's going to equal,
parentheses, logarithm--

00:08:56.470 --> 00:08:57.580
oh, I forgot my base.

00:08:57.580 --> 00:09:02.490
Logarithm base 2 of
32 minus, right?

00:09:02.490 --> 00:09:04.010
Because this is in a quotient.

00:09:04.010 --> 00:09:11.320
Minus the logarithm base 2
of the square root of 8.

00:09:11.320 --> 00:09:12.660
Right?

00:09:12.660 --> 00:09:13.410
Let's see.

00:09:13.410 --> 00:09:16.070
Well here once again we have a
square root here, so we could

00:09:16.070 --> 00:09:22.190
say this is equal to 1/2
times log base 2 of 32.

00:09:22.190 --> 00:09:25.200
Minus this 8 to the 1/2,
which is the same thing

00:09:25.200 --> 00:09:28.860
is 1/2 log base 2 of 8.

00:09:28.860 --> 00:09:31.390
We learned that property in the
beginning of this presentation.

00:09:31.390 --> 00:09:33.750
And then if we want, we can
distribute this original 1/2.

00:09:33.750 --> 00:09:42.220
This equals 1/2 log base 2 of
32 minus 1/4-- because we have

00:09:42.220 --> 00:09:46.960
to distribute that 1/2--
minus 1/4 log base 2 of 8.

00:09:46.960 --> 00:09:52.500
This is 5/2 minus, this is 3.

00:09:52.500 --> 00:09:55.120
3 times 1/4 minus 3/4.

00:09:55.120 --> 00:09:59.440
Or 10/4 minus 3/4
is equal to 7/4.

00:09:59.440 --> 00:10:03.150
I probably made some arithmetic
errors, but you get the point.

00:10:03.150 --> 00:10:04.730
See you soon!


Introduction to logarithm properties

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:03.585
Welcome to this presentation
on logarithm properties.

00:00:03.585 --> 00:00:05.970
Now this is going to be a
very hands-on presentation.

00:00:05.970 --> 00:00:08.580
If you don't believe that one
of these properties are true

00:00:08.580 --> 00:00:11.840
and you want them proved, I've
made three or four videos

00:00:11.840 --> 00:00:12.930
that actually prove
these properties.

00:00:12.930 --> 00:00:14.000
But what I'm going to do
is I'm going to show

00:00:14.000 --> 00:00:14.620
you the properties.

00:00:14.620 --> 00:00:15.900
And then show you how
they can be used.

00:00:15.900 --> 00:00:17.520
It's going to be
little more hands-on.

00:00:17.520 --> 00:00:20.770
So let's just do a little
bit of a review of just

00:00:20.770 --> 00:00:23.000
what a logarithm is.

00:00:23.000 --> 00:00:28.540
So if I say that a-- Oh
that's not the right.

00:00:28.540 --> 00:00:28.930
Let's see.

00:00:28.930 --> 00:00:31.950
I want to change--
There you go.

00:00:31.950 --> 00:00:35.260
Let's say I say that
a-- Let me start over.

00:00:35.260 --> 00:00:41.710
a to the b is equal to c.

00:00:41.710 --> 00:00:44.660
So if we-- a to the b
power is equal to c.

00:00:44.660 --> 00:00:47.420
So another way to write this
exact same relationship instead

00:00:47.420 --> 00:00:49.970
of writing the exponent, is
to write it as a logarithm.

00:00:49.970 --> 00:00:57.550
So we can say that the
logarithm base a of

00:00:57.550 --> 00:01:02.040
c is equal to b.

00:01:02.040 --> 00:01:05.000
So these are essentially
saying the same thing.

00:01:05.000 --> 00:01:06.750
They just have different
kind of results.

00:01:06.750 --> 00:01:09.710
In one, you know a and b and
you're kind of getting c.

00:01:09.710 --> 00:01:12.020
That's what exponentiation
does for you.

00:01:12.020 --> 00:01:14.250
And the second one, you know
a and you know that when

00:01:14.250 --> 00:01:16.340
you raise it to some
power you get c.

00:01:16.340 --> 00:01:18.040
And then you figure
out what b is.

00:01:18.040 --> 00:01:20.540
So they're the exact same
relationship, just stated

00:01:20.540 --> 00:01:22.090
in a different way.

00:01:22.090 --> 00:01:24.910
Now I will introduce you
to some interesting

00:01:24.910 --> 00:01:25.970
logarithm properties.

00:01:25.970 --> 00:01:30.480
And they actually just fall
out of this relationship and

00:01:30.480 --> 00:01:32.740
the regular exponent rules.

00:01:32.740 --> 00:01:36.865
So the first is that the
logarithm-- Let me do

00:01:36.865 --> 00:01:39.350
a more cheerful color.

00:01:39.350 --> 00:01:45.260
The logarithm, let's say, of
any base-- So let's just call

00:01:45.260 --> 00:01:47.190
the base-- Let's
say b for base.

00:01:47.190 --> 00:01:57.820
Logarithm base b of a plus
logarithm base b of c-- and

00:01:57.820 --> 00:01:59.770
this only works if we
have the same bases.

00:01:59.770 --> 00:02:01.800
So that's important
to remember.

00:02:01.800 --> 00:02:12.790
That equals the logarithm
of base b of a times c.

00:02:12.790 --> 00:02:14.660
Now what does this mean
and how can we use it?

00:02:14.660 --> 00:02:18.320
Or let's just even try it
out with some, well I

00:02:18.320 --> 00:02:19.630
don't know, examples.

00:02:19.630 --> 00:02:22.860
So this is saying that-- I'll
switch to another color.

00:02:22.860 --> 00:02:25.390
Let's make mauve my--
Mauve-- I don't know.

00:02:25.390 --> 00:02:26.730
I never know how to
say that properly.

00:02:26.730 --> 00:02:29.470
Let's make that my
example color.

00:02:29.470 --> 00:02:40.920
So let's say logarithm of base
2 of-- I don't know --of 8

00:02:40.920 --> 00:02:50.570
plus logarithm base 2 of-- I
don't know let's say --32.

00:02:53.210 --> 00:02:58.410
So, in theory, this should
equal, if we believe this

00:02:58.410 --> 00:03:05.500
property, this should equal
logarithm base 2 of what?

00:03:05.500 --> 00:03:07.910
Well we say 8 times 32.

00:03:07.910 --> 00:03:17.660
So 8 times 32 is
240 plus 16, 256.

00:03:17.660 --> 00:03:18.510
Well let's see if that's true.

00:03:18.510 --> 00:03:20.740
Just trying out this number and
this is really isn't a proof.

00:03:20.740 --> 00:03:22.860
But it'll give you a little bit
of an intuition, I think, for

00:03:22.860 --> 00:03:23.990
what's going on around you.

00:03:23.990 --> 00:03:26.340
So log-- So this is-- We
just used our property.

00:03:26.340 --> 00:03:28.480
This little property that
I presented to you.

00:03:28.480 --> 00:03:29.890
And let's just see
if it works out.

00:03:29.890 --> 00:03:31.860
So log base 2 of 8.

00:03:31.860 --> 00:03:34.520
2 to what power is equal to 8?

00:03:34.520 --> 00:03:38.960
Well 2 to the third power
is equal to 8, right?

00:03:38.960 --> 00:03:41.310
So this term right here,
that equals 3, right?

00:03:41.310 --> 00:03:44.810
Log base 2 of 8 is equal to 3.

00:03:44.810 --> 00:03:48.400
2 to what power is equal to 32?

00:03:48.400 --> 00:03:48.790
Let's see.

00:03:48.790 --> 00:03:50.580
2 to the fourth power is 16.

00:03:50.580 --> 00:03:53.300
2 to the fifth power is 32.

00:03:53.300 --> 00:03:58.410
So this right here is 2 to
the-- This is 5, right?

00:03:58.410 --> 00:04:02.750
And 2 to the what power
is equal to 256?

00:04:02.750 --> 00:04:05.890
Well if you're a computer
science major, you'll

00:04:05.890 --> 00:04:07.210
know that immediately.

00:04:07.210 --> 00:04:10.470
That a byte can have
256 values in it.

00:04:10.470 --> 00:04:12.440
So it's 2 to the eighth power.

00:04:12.440 --> 00:04:15.520
But if you don't know that, you
could multiply it out yourself.

00:04:15.520 --> 00:04:16.710
But this is 8.

00:04:16.710 --> 00:04:18.495
And I'm not doing it just
because I knew that 3

00:04:18.495 --> 00:04:19.470
plus 5 is equal to 8.

00:04:19.470 --> 00:04:21.230
I'm doing this independently.

00:04:21.230 --> 00:04:22.300
So this is equal to 8.

00:04:22.300 --> 00:04:28.530
But it does turn out that
3 plus 5 is equal to 8.

00:04:28.530 --> 00:04:32.170
This may seem like magic to
you or it may seem obvious.

00:04:32.170 --> 00:04:35.560
And for those of you who it
might seem a little obvious,

00:04:35.560 --> 00:04:43.320
you're probably thinking, well
2 to the third times 2 to the

00:04:43.320 --> 00:04:49.510
fifth is equal to 2 to
the 3 plus 5, right?

00:04:49.510 --> 00:04:51.800
This is just an exponent rule.

00:04:51.800 --> 00:04:52.670
What do they call this?

00:04:52.670 --> 00:04:54.750
The additive exponent
prop-- I don't know.

00:04:54.750 --> 00:04:56.220
I don't know the
names of things.

00:04:56.220 --> 00:04:59.920
And that equals 2 to
8, 2 to the eighth.

00:04:59.920 --> 00:05:03.160
And that's exactly what
we did here, right?

00:05:03.160 --> 00:05:05.830
On this side, we had 2
the third times 2 to

00:05:05.830 --> 00:05:07.000
the fifth, essentially.

00:05:07.000 --> 00:05:09.780
And on this side, you have
them added to each other.

00:05:09.780 --> 00:05:12.850
And what makes the logarithms
interesting is and why-- It's

00:05:12.850 --> 00:05:13.970
a little confusing at first.

00:05:13.970 --> 00:05:15.620
And you can watch the proofs
if you really want kind of

00:05:15.620 --> 00:05:18.450
a rigorous-- my proofs
aren't rigorous.

00:05:18.450 --> 00:05:19.940
But if you want kind of
a better explanation

00:05:19.940 --> 00:05:21.360
of how this works.

00:05:21.360 --> 00:05:23.480
But this should hopefully give
you an tuition for why this

00:05:23.480 --> 00:05:25.330
property holds, right?

00:05:25.330 --> 00:05:27.330
Because when you multiply
two numbers of the

00:05:27.330 --> 00:05:29.270
same base, right?

00:05:29.270 --> 00:05:31.720
Two exponential expressions
of the same base, you

00:05:31.720 --> 00:05:33.560
can add their exponents.

00:05:33.560 --> 00:05:37.050
Similarly, when you have the
log of two numbers multiplied

00:05:37.050 --> 00:05:41.710
by each other, that's
equivalent to the log of each

00:05:41.710 --> 00:05:43.530
of the numbers added
to each other.

00:05:43.530 --> 00:05:45.810
This is the same property.

00:05:45.810 --> 00:05:49.940
If you don't believe me,
watch the proof videos.

00:05:49.940 --> 00:05:56.380
So let's do a-- Let me show
you another log property.

00:05:56.380 --> 00:05:58.230
It's pretty much the same one.

00:05:58.230 --> 00:05:59.400
I almost view them the same.

00:05:59.400 --> 00:06:10.450
So this is log base b of
a minus log base b of c

00:06:10.450 --> 00:06:17.040
is equal to log base b
of-- well I ran out.

00:06:17.040 --> 00:06:19.290
I'm running out of space
--a divided by c.

00:06:19.290 --> 00:06:21.540
That says a divided by c.

00:06:21.540 --> 00:06:25.240
And we can, once again, try
it out with some numbers.

00:06:25.240 --> 00:06:28.620
I use 2 a lot just because
2 is an easy number to

00:06:28.620 --> 00:06:29.790
figure out the powers.

00:06:29.790 --> 00:06:30.750
But let's use a
different number.

00:06:30.750 --> 00:06:41.200
Let's say log base 3 of-- I
don't know --log base 3 of--

00:06:41.200 --> 00:06:44.820
well you know, let's make it
interesting --log base 3 of

00:06:44.820 --> 00:06:56.930
1/9 minus log base 3 of 81.

00:06:56.930 --> 00:07:02.500
So this property tells us--
This is the same thing as--

00:07:02.500 --> 00:07:04.180
Well I'm ending up
with a big number.

00:07:04.180 --> 00:07:12.990
Log base 3 of 1/9
divided by 81.

00:07:12.990 --> 00:07:15.890
So that's the same thing
as 1/9 times 1/81.

00:07:15.890 --> 00:07:19.710
I used two large numbers
for my example, but

00:07:19.710 --> 00:07:21.320
we'll move forward.

00:07:21.320 --> 00:07:21.750
So let's see.

00:07:21.750 --> 00:07:25.700
9 times 8 is 720, right?

00:07:25.700 --> 00:07:27.190
9 times-- Right.

00:07:27.190 --> 00:07:28.580
9 times 8 is 720.

00:07:28.580 --> 00:07:31.036
So this is 1/729.

00:07:31.036 --> 00:07:37.860
So this is log base
3 over 1/729.

00:07:37.860 --> 00:07:42.160
So what-- What does-- 3 to
what power is equal to 1/9?

00:07:42.160 --> 00:07:45.340
Well 3 squared is
equal to 9, right?

00:07:48.070 --> 00:07:53.250
So 3-- So we know that if 3
squared is equal to 9, then we

00:07:53.250 --> 00:07:56.950
know that 3 to the negative
2 is equal to 1/9, right?

00:07:56.950 --> 00:07:58.190
The negative just inverts it.

00:07:58.190 --> 00:08:02.160
So this is equal to
negative 2, right?

00:08:02.160 --> 00:08:06.120
And then minus-- 3 to
what power is equal 81?

00:08:06.120 --> 00:08:07.870
3 to the third power is 27.

00:08:07.870 --> 00:08:11.390
So 3 to the fourth power.

00:08:11.390 --> 00:08:15.870
So we have minus 2 minus 4 is
equal to-- Well, we could

00:08:15.870 --> 00:08:16.710
do it a couple of ways.

00:08:16.710 --> 00:08:20.530
Minus 2 minus 4 is
equal to minus 6.

00:08:20.530 --> 00:08:23.280
And now we just have to confirm
that 3 to the minus sixth

00:08:23.280 --> 00:08:26.080
power is equal to 1/729.

00:08:26.080 --> 00:08:26.890
So that's my question.

00:08:26.890 --> 00:08:34.430
Is 3 to the minus sixth power,
is that equal to 7-- 1/729?

00:08:34.430 --> 00:08:36.900
Well that's the same thing as
saying 3 to sixth power is

00:08:36.900 --> 00:08:40.110
equal to 729, because that's
all the negative exponent

00:08:40.110 --> 00:08:41.760
does is inverts it.

00:08:41.760 --> 00:08:42.590
Let's see.

00:08:42.590 --> 00:08:44.880
We could multiply that out,
but that should be the case.

00:08:44.880 --> 00:08:46.480
Because, well, we
could look here.

00:08:46.480 --> 00:08:47.320
But let's see.

00:08:47.320 --> 00:08:52.630
3 to the third power-- This
would be 3 to the third power

00:08:52.630 --> 00:08:57.450
times 3 to the third power
is equal to 27 times 27.

00:08:57.450 --> 00:08:58.540
That looks pretty close.

00:08:58.540 --> 00:09:00.900
You can confirm it with
a calculator if you

00:09:00.900 --> 00:09:02.340
don't believe me.

00:09:02.340 --> 00:09:04.940
Anyway, that's all the time
I have in this video.

00:09:04.940 --> 00:09:07.500
In the next video, I'll
introduce you to the last

00:09:07.500 --> 00:09:08.680
two logarithm properties.

00:09:08.680 --> 00:09:12.490
And, if we have time, maybe
I'll do examples with

00:09:12.490 --> 00:09:13.440
the leftover time.

00:09:13.440 --> 00:09:15.120
I'll see you soon.


Proof: log_a (B) = (log_x (B))/(log_x (A))

WEBVTT
Kind: captions
Language: en

00:00:00.860 --> 00:00:04.390
I have one last -- I was going
to say, trick property.

00:00:04.390 --> 00:00:07.150
One last logarithm
property to show you.

00:00:07.150 --> 00:00:10.390
So let me pick a suitably
festive color for

00:00:10.390 --> 00:00:11.950
this last property.

00:00:11.950 --> 00:00:19.620
So let's say that just, I don't
know, x to the n is equal to a.

00:00:19.620 --> 00:00:20.720
Nothing fancy there.

00:00:20.720 --> 00:00:29.510
Well, that's just another way
of saying that log base x

00:00:29.510 --> 00:00:31.720
of a is equal to n, right?

00:00:31.720 --> 00:00:36.180
That's the exact same -- this
is just the exact same way of

00:00:36.180 --> 00:00:38.150
writing the exact same thing.

00:00:38.150 --> 00:00:40.410
One's a logarithm, one's
an exponent, right?

00:00:40.410 --> 00:00:43.240
These imply the same thing.

00:00:43.240 --> 00:00:45.890
But what we can do is, if n is
actually equal to this

00:00:45.890 --> 00:00:49.140
expression, we can, like I did
a couple of videos ago, you

00:00:49.140 --> 00:00:51.150
could just substitute
this for n.

00:00:51.150 --> 00:01:01.220
So we could write x to
this thing, log base x a.

00:01:01.220 --> 00:01:04.470
And we could set that
as equal to what?

00:01:04.470 --> 00:01:06.950
a.

00:01:06.950 --> 00:01:08.260
Fascinating.

00:01:08.260 --> 00:01:10.980
So now what I'm going to do
and, actually, this is going to

00:01:10.980 --> 00:01:17.000
get pretty messy is, I'm going
to raise -- actually, let me

00:01:17.000 --> 00:01:19.480
write this a little more space.

00:01:22.360 --> 00:01:22.685
Undo.

00:01:28.060 --> 00:01:29.550
Oh I can't keep undoing.

00:01:29.550 --> 00:01:32.260
Anyway, so let me write
down here with more space.

00:01:32.260 --> 00:01:34.100
Because I'm going to
do something fancy.

00:01:34.100 --> 00:01:35.870
So, ignore this.

00:01:35.870 --> 00:01:47.350
So, if I set x to the log base
x of a, that equals -- and

00:01:47.350 --> 00:01:50.060
you'll see why I'm giving you
so much space right now.

00:01:50.060 --> 00:01:52.100
Equals a.

00:01:52.100 --> 00:01:54.540
Now, what I want to do is,
I want to raise both sides

00:01:54.540 --> 00:01:58.690
of this equation to 1
over this exponent.

00:01:58.690 --> 00:02:05.320
So I'm going to raise that
to 1 over log base x of a.

00:02:05.320 --> 00:02:07.060
If I do something to one side
of the equation, I have

00:02:07.060 --> 00:02:08.520
to do it to the other.

00:02:08.520 --> 00:02:14.490
So that's also, that's equal to
a, to 1 over log base x to a.

00:02:14.490 --> 00:02:17.590
I know, this is quite
daunting already.

00:02:17.590 --> 00:02:18.490
But you'll see where I'm going.

00:02:18.490 --> 00:02:22.140
And hopefully nothing
I've done is completely

00:02:22.140 --> 00:02:24.750
not-intuitive, right?

00:02:24.750 --> 00:02:28.500
This expression is just another
way of writing this expression.

00:02:28.500 --> 00:02:29.680
And I substitut it for n.

00:02:29.680 --> 00:02:31.800
And now I'm raising
both to this exponent.

00:02:31.800 --> 00:02:33.280
And you'll see why
I'm doing that.

00:02:33.280 --> 00:02:34.870
Well, if you're raising
something to an the exponent

00:02:34.870 --> 00:02:37.000
and then you're raising that
to an exponent, you just

00:02:37.000 --> 00:02:38.940
multiply the two, right?

00:02:38.940 --> 00:02:40.340
So they cancel out.

00:02:40.340 --> 00:02:41.655
Because this will
be the numerator.

00:02:41.655 --> 00:02:43.600
And this'll be the denominator.

00:02:43.600 --> 00:02:50.140
So that gets us to this.

00:02:50.140 --> 00:02:51.750
x to the 1 power, right?

00:02:51.750 --> 00:02:55.610
Because log base x of a over
log base x of a is equal to 1.

00:02:55.610 --> 00:03:02.100
So that's the same thing
as x is equal to a to the

00:03:02.100 --> 00:03:08.460
1 over log base x of a.

00:03:08.460 --> 00:03:10.770
You're probably saying, Sal,
where are you going with this.

00:03:10.770 --> 00:03:13.300
And I will sort
show you shortly.

00:03:13.300 --> 00:03:17.280
So, we could also just replace
a with another variable, right?

00:03:17.280 --> 00:03:23.660
I could also write x is also
equal to b to the 1 over

00:03:23.660 --> 00:03:27.980
log base x of b, right?

00:03:27.980 --> 00:03:30.110
Nothing strange there.

00:03:30.110 --> 00:03:32.990
The same exact thing I did
with a, I could do with a.

00:03:32.990 --> 00:03:35.350
The same thing I did with
a, I could do with b.

00:03:35.350 --> 00:03:37.840
So I've written these
two expressions.

00:03:37.840 --> 00:03:40.020
I said x is equal to
both of these things.

00:03:40.020 --> 00:03:42.020
So let's set them
equal to each other.

00:03:42.020 --> 00:03:55.230
So, we know that a to 1 over
log base x of a, is equal to b

00:03:55.230 --> 00:04:00.370
to the 1 over log base x of b.

00:04:03.010 --> 00:04:04.730
So, what can we do now?

00:04:04.730 --> 00:04:07.200
Well let's raise both of these
-- actually, I'm running

00:04:07.200 --> 00:04:08.220
out of so much space.

00:04:08.220 --> 00:04:11.060
Let me clear this and go
to the next page, or

00:04:11.060 --> 00:04:12.260
go to another page.

00:04:14.770 --> 00:04:16.040
Clear image.

00:04:16.040 --> 00:04:16.610
Invert.

00:04:16.610 --> 00:04:17.960
So what did I just write?

00:04:17.960 --> 00:04:19.590
I said that, because I
need a lot of space

00:04:19.590 --> 00:04:21.860
for what I plan to do.

00:04:21.860 --> 00:04:33.050
So, I said, a to the 1 over log
base x of a -- well, that

00:04:33.050 --> 00:04:41.380
equals b to the 1 over
log base x of b.

00:04:41.380 --> 00:04:43.640
And hopefully you're
satisfied with that.

00:04:43.640 --> 00:04:48.980
Now, let's raise both of
these sides to the log

00:04:48.980 --> 00:04:50.820
base x of b power.

00:04:53.620 --> 00:04:57.100
This long base x's of b power.

00:04:57.100 --> 00:04:58.800
Now, hopefully you'll
see why I'm doing this.

00:04:58.800 --> 00:05:00.400
On this side they'll
cancel out, right?

00:05:00.400 --> 00:05:01.520
Because this becomes
a numerator, that's

00:05:01.520 --> 00:05:02.630
the denominator.

00:05:02.630 --> 00:05:10.260
And on this side, you get a
to the -- this becomes the

00:05:10.260 --> 00:05:13.420
numerator, right, because we
just multiply the exponents.

00:05:13.420 --> 00:05:18.440
Log base x, that
little dot is an x.

00:05:18.440 --> 00:05:26.520
Of b over log base x of a.

00:05:26.520 --> 00:05:28.370
And what does that equal?

00:05:28.370 --> 00:05:30.080
Well, that equals
just b, right?

00:05:30.080 --> 00:05:31.580
Because this over this is 1.

00:05:31.580 --> 00:05:32.890
This b to the 1.

00:05:32.890 --> 00:05:34.920
That equals b.

00:05:34.920 --> 00:05:39.860
Now let's write this entire
thing as a logarithm.

00:05:39.860 --> 00:05:42.710
a to this thing is equal to b.

00:05:42.710 --> 00:05:49.840
That's the exact same thing as
saying that the logarithm base

00:05:49.840 --> 00:05:55.730
a of b is equal to this thing.

00:05:55.730 --> 00:06:05.350
Is equal to the log base x of b
divided by the log base x of a.

00:06:05.350 --> 00:06:08.890
This might seem confusing, it
might seem daunting, but we're

00:06:08.890 --> 00:06:10.020
actually going to do a lot
of examples with this.

00:06:10.020 --> 00:06:14.780
And this is probably the single
most useful identity, I

00:06:14.780 --> 00:06:17.700
guess you could call it, if
you're using a calculator.

00:06:17.700 --> 00:06:18.490
Why?

00:06:18.490 --> 00:06:20.135
Because your calculator
only has two bases.

00:06:20.135 --> 00:06:27.460
It either has log base, you
know base 10, or base e, right?

00:06:27.460 --> 00:06:29.280
And most of them, when
you press the log button

00:06:29.280 --> 00:06:32.390
on your calculator, it
assumes log base 10.

00:06:32.390 --> 00:06:40.170
So if I gave you a problem
where I wanted to know what is

00:06:40.170 --> 00:06:43.580
the log base 7 of 3, right?

00:06:43.580 --> 00:06:44.130
Who knows?

00:06:44.130 --> 00:06:45.620
7 to what power is 3?

00:06:45.620 --> 00:06:49.380
And there's no easy way, on
most calculators, to do this.

00:06:49.380 --> 00:06:50.520
Well, you can use
this identity.

00:06:50.520 --> 00:06:57.150
That this is the same thing as
the log base 10 of 3, divided

00:06:57.150 --> 00:07:00.530
by the log base 10 of 7.

00:07:00.530 --> 00:07:03.440
And these are very easy to
calculate on your calculator.

00:07:03.440 --> 00:07:05.130
You just type 3 and press log.

00:07:05.130 --> 00:07:05.970
It'll give you this number.

00:07:05.970 --> 00:07:08.170
And you press 7 and click
on log, it'll give

00:07:08.170 --> 00:07:08.930
you this number.

00:07:08.930 --> 00:07:10.120
And then you're done.

00:07:10.120 --> 00:07:13.230
So hopefully you're satisfied
that this is true and you

00:07:13.230 --> 00:07:14.940
have a little bit of an
intuition of how to use it.

00:07:14.940 --> 00:07:17.920
And I'll make a bunch of videos
now, on actually how you can

00:07:17.920 --> 00:07:19.280
use these logarithm properties.

00:07:19.280 --> 00:07:21.140
I just wanted to get it out
of the way so that you're

00:07:21.140 --> 00:07:24.450
satisfied that they are true.

00:07:24.450 --> 00:07:26.230
I'll see you soon.