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We're asked to graph, y is
equal to log base 5 of x.

3
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And just to remind us
what this is saying,

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this is saying that y
is equal to the power

5
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that I have to raise
5 to to get to x.

6
00:00:13,460 --> 00:00:15,980
Or if I were to write
this logarithmic equation

7
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as an exponential
equation, 5 is my base,

8
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y is the exponent that I
have to raise my base to,

9
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and then x is what I get when
I raise 5 to the yth power.

10
00:00:28,420 --> 00:00:32,570
So another way of writing
this equation would be 5

11
00:00:32,570 --> 00:00:40,710
to the y'th power is
going to be equal to x.

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These are the same thing.

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Here, we have y as
a function of x.

14
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Here, we have x as
a function of y.

15
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But they're really saying
the exact same thing,

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raise 5 to the y'th
power to get x.

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When you put it as a
logarithm, you're saying, well,

18
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what power do I have
to raise 5 to to get x?

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We'll have to raise it to y.

20
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Here, what do I get when I
raise five to the y power?

21
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I get x.

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That out of the way, let's
make ourselves a little table

23
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that we can use to
plot some points,

24
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and then we can
connect the dots to see

25
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what this curve looks like.

26
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So let me pick some
x's and some y's.

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And we, in general, want to
pick some numbers that give us

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some nice round answers, some
nice fairly simple numbers

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for us to deal with,
so that we don't

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have to get the calculator.

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And so in general,
you want to pick

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x values where the power
that you have to raise 5

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00:01:34,250 --> 00:01:38,156
to to get that x value is a
pretty straightforward power.

35
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Or another way to
think about it,

36
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you could just think about
the different y values

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that you want to raise
5 to the power of,

38
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and then you could
get your x values.

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So we could actually
think about this one

40
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to come up with our
actual x values.

41
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But we want to be clear that
when we express it like this,

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the independent variable is x,
and the dependent variable is

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y.

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We might just look at
this one to pick some nice

45
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even or nice x's that give
us nice clean answers for y.

46
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So here, I'm actually going
to fill in the y first,

47
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just so we get nice clean x's.

48
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So let's say we're
going to raise five

49
00:02:16,500 --> 00:02:19,120
to the-- let's say we're going
to raise it-- I'm going to pick

50
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some new colors-- negative
2, negative 2 power--

51
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and let me do some other
colors-- negative 1, 0, 1.

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I'll do one more, and then 2.

53
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So once again, this is
a little nontraditional,

54
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where I'm filling in the
dependent variable first.

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But the way that
we've written it over

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here, it's actually given
the dependent variable,

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it's easy to figure out what
the independent variable needs

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to be for this
logarithmic function.

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So, what x gives me
a y of negative 2?

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What x gives me--
what does x have

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to be for y to be
equal to negative 2?

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Well, 5 to the negative 2 power
is going to be equal to x.

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So 5 to the negative
2 is 1 over 25.

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So we get 1 over 25.

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If we go back to
this earlier one,

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if we say log base
5 of 1 over 25,

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what power do I have to
raise 5 to to get 1 over 25?

68
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We'll have to raise it
to the negative 2 power.

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Or you could say 5
to the negative 2

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is equal to 1 over 25.

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These are saying the
exact same thing.

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Now let's do another one.

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What happens when I raise
5 to the negative 1 power?

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I get one fifth.

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So if we go to this
original one over there,

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we're just saying that
log base 5 of one fifth.

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Want to be careful.

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This is saying, what power
do I have to raise 5 to

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in order to get one fifth.

80
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We'll have to raise it
to the negative 1 power.

81
00:03:50,640 --> 00:03:53,150


82
00:03:53,150 --> 00:03:55,080
What happens when I take
5 to the 0'th power?

83
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I get one.

84
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And so this relationship--
This is the same thing

85
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as saying log base 5 of 1.

86
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What power do I have
to raise 5 to to get 1?

87
00:04:05,430 --> 00:04:08,910
I just have to raise
it to the 0th power.

88
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Let's do the next two.

89
00:04:10,660 --> 00:04:13,470
What happens when I raise
5 to the first power?

90
00:04:13,470 --> 00:04:17,180
Well, I get 5 So if you go look
over here, that's just saying,

91
00:04:17,180 --> 00:04:20,410
log, what power do I have
to raise 5 to to get 5?

92
00:04:20,410 --> 00:04:23,880
We'll have to just raise
it to the first power.

93
00:04:23,880 --> 00:04:28,800
And then finally, if I
take 5 squared, I get 25.

94
00:04:28,800 --> 00:04:31,680
So when you look at it from
the logarithmic point of view,

95
00:04:31,680 --> 00:04:34,410
you say, well, what power
do I have to raise 5 to

96
00:04:34,410 --> 00:04:36,020
to get to 25?

97
00:04:36,020 --> 00:04:38,930
We'll have to raise it
to the second power.

98
00:04:38,930 --> 00:04:41,830
So I took the inverse of
the logarithmic function.

99
00:04:41,830 --> 00:04:43,550
I wrote it as an
exponential function.

100
00:04:43,550 --> 00:04:47,270
I switched the dependent
and independent variables,

101
00:04:47,270 --> 00:04:50,760
so I can derive nice clean x's
that will give me nice clean

102
00:04:50,760 --> 00:04:51,734
y's.

103
00:04:51,734 --> 00:04:54,150
Now with that out of the way,
but I do want to remind you,

104
00:04:54,150 --> 00:04:57,497
I could have just picked
random numbers over here,

105
00:04:57,497 --> 00:04:59,830
but then I would have probably
gotten less clean numbers

106
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over here.

107
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I would have had to
use a calculator.

108
00:05:01,510 --> 00:05:03,093
The only reason why
I did it this way,

109
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is so I get nice clean results
that I can plot by hand.

110
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So let's actually graph it.

111
00:05:08,740 --> 00:05:10,910
Let's actually graph
this thing over here.

112
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So the y's go between
negative 2 and 2.

113
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The x's go from 1/25th
all the way to 25.

114
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So let's graph it.

115
00:05:22,680 --> 00:05:30,050
So that is my y-axis,
and this is my x-axis.

116
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Draw it like that.

117
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That is my x-axis.

118
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And then the y's start at 0.

119
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Then, you get to
positive 1, positive 2.

120
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And then you have negative 1.

121
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And you have negative 2.

122
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And then on the x-axis,
it's all positive.

123
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And I'll let you think about
whether the domain here

124
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is-- well, when you
think about it--

125
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is a logarithmic
function defined

126
00:05:58,270 --> 00:06:03,050
for an x that is not positive?

127
00:06:03,050 --> 00:06:07,190
So is there any power that I can
raise five to that I can get 0?

128
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No.

129
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You could raise five to an
infinitely negative power

130
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to get a very, very, very, very
small number that approaches

131
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zero, but you can
never get-- there's

132
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no power that you can
raise 5 to to get 0.

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So x cannot be 0.

134
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And there's no power
then you could raise 5

135
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to get another negative number.

136
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So x can also not be
a negative number.

137
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So the domain of this function
right over here-- and this

138
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is relevant, because we want
to think about what we're

139
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graphing-- the domain here is
x has to be greater than zero.

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Let me write that down.

141
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The domain here is that x
has to be greater than 0.

142
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So we're only going
to be able to graph

143
00:06:41,300 --> 00:06:45,660
this function in
the positive x-axis.

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So with that out of the
way, x gets as large as 25.

145
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So let me graph-- we
put those points here.

146
00:06:51,150 --> 00:06:57,430
So that is 5, 10,
15, 20, and 25.

147
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And then let's plot these.

148
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So the first one is in blue.

149
00:07:00,050 --> 00:07:02,720
When x is 1/25 and
y is negative 2--

150
00:07:02,720 --> 00:07:06,090
When x is 1/25 so
1 is there-- 1/25

151
00:07:06,090 --> 00:07:09,940
is going to be really close to
there-- Then y is negative 2.

152
00:07:09,940 --> 00:07:12,680
So it's going to be
like right over there,

153
00:07:12,680 --> 00:07:14,190
not quite at the y-axis.

154
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We're at 1/25 to the
right of the y-axis.

155
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But pretty close.

156
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So that's right over there.

157
00:07:19,120 --> 00:07:23,680
That is 1 over 25, comma
negative 2 right over there.

158
00:07:23,680 --> 00:07:26,270
Then, when x is one
fifth, which is slightly

159
00:07:26,270 --> 00:07:30,550
further to the right, one
fifth y is negative 1.

160
00:07:30,550 --> 00:07:32,880
So right over there.

161
00:07:32,880 --> 00:07:36,950
So this is one
fifth, negative 1.

162
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Then when x is 1, y is 0.

163
00:07:40,290 --> 00:07:43,310
So 1 might be right there.

164
00:07:43,310 --> 00:07:46,550
So this is the point 1,0.

165
00:07:46,550 --> 00:07:50,630
And then when x is 5, y is 1.

166
00:07:50,630 --> 00:07:53,180
When x is 5, I covered it
over here, when this is five,

167
00:07:53,180 --> 00:07:56,530
y is 1.

168
00:07:56,530 --> 00:07:59,280
So that's the point 5,1.

169
00:07:59,280 --> 00:08:02,200
And then finally,
when x is 25, y is 2.

170
00:08:02,200 --> 00:08:08,060


171
00:08:08,060 --> 00:08:11,140
So this is 25,2.

172
00:08:11,140 --> 00:08:13,050
And then I can
graph the function.

173
00:08:13,050 --> 00:08:17,030
And I'll do it-- let me do it
in a color-- I'll use this pink.

174
00:08:17,030 --> 00:08:23,550
So as x gets super, super,
super, super small, y goes

175
00:08:23,550 --> 00:08:25,820
to negative infinity.

176
00:08:25,820 --> 00:08:30,490
It gets really small-- to
get x's or as x becomes--

177
00:08:30,490 --> 00:08:34,179
if you say what power do
you have to raise 5 to

178
00:08:34,179 --> 00:08:36,539
to get 0.0001?

179
00:08:36,539 --> 00:08:38,530
It has to be very, very,
very negative power.

180
00:08:38,530 --> 00:08:43,090
So y is going to be very
negative as we approach 0.

181
00:08:43,090 --> 00:08:47,390
And then it kind of
moves up like that.

182
00:08:47,390 --> 00:08:53,110
And then starts to kind of
curve to the right like that.

183
00:08:53,110 --> 00:08:54,840
And this thing
right over here, is

184
00:08:54,840 --> 00:08:58,890
going to keep going down at
a steeper and steeper rate.

185
00:08:58,890 --> 00:09:03,160
And it's never going
to quite touch.

186
00:09:03,160 --> 00:09:04,060
the y-axis.

187
00:09:04,060 --> 00:09:06,370
It's going to get closer
and closer to the y-axis.

188
00:09:06,370 --> 00:09:09,840
But it's never going
to be quite touch it.