WEBVTT

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

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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

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

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Or if I were to write
this logarithmic equation

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as an exponential
equation, 5 is my base,

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

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

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So another way of writing
this equation would be 5

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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.

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Here, we have x as
a function of y.

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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,

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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.

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

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I get x.

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

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that we can use to
plot some points,

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and then we can
connect the dots to see

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what this curve looks like.

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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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to to get that x value is a
pretty straightforward power.

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Or another way to
think about it,

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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,

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and then you could
get your x values.

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So we could actually
think about this one

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to come up with our
actual x values.

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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

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

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

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just so we get nice clean x's.

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

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to the-- let's say we're going
to raise it-- I'm going to pick

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

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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.

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So once again, this is
a little nontraditional,

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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?

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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.

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

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What happens when I take
5 to the 0'th power?

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I get one.

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And so this relationship--
This is the same thing

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as saying log base 5 of 1.

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

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I just have to raise
it to the 0th power.

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Let's do the next two.

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What happens when I raise
5 to the first power?

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Well, I get 5 So if you go look
over here, that's just saying,

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log, what power do I have
to raise 5 to to get 5?

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We'll have to just raise
it to the first power.

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And then finally, if I
take 5 squared, I get 25.

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So when you look at it from
the logarithmic point of view,

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you say, well, what power
do I have to raise 5 to

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to get to 25?

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We'll have to raise it
to the second power.

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So I took the inverse of
the logarithmic function.

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I wrote it as an
exponential function.

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I switched the dependent
and independent variables,

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so I can derive nice clean x's
that will give me nice clean

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y's.

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Now with that out of the way,
but I do want to remind you,

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I could have just picked
random numbers over here,

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but then I would have probably
gotten less clean numbers

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over here.

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I would have had to
use a calculator.

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The only reason why
I did it this way,

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

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So let's actually graph it.

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Let's actually graph
this thing over here.

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

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

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So let's graph it.

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So that is my y-axis,
and this is my x-axis.

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Draw it like that.

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That is my x-axis.

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And then the y's start at 0.

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Then, you get to
positive 1, positive 2.

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And then you have negative 1.

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And you have negative 2.

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

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

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

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is a logarithmic
function defined

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for an x that is not positive?

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So is there any power that I can
raise five to that I can get 0?

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No.

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You could raise five to an
infinitely negative power

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

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

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

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So x cannot be 0.

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And there's no power
then you could raise 5

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to get another negative number.

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So x can also not be
a negative number.

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

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

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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.

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

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So we're only going
to be able to graph

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this function in
the positive x-axis.

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

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

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So that is 5, 10,
15, 20, and 25.

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And then let's plot these.

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So the first one is in blue.

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When x is 1/25 and
y is negative 2--

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When x is 1/25 so
1 is there-- 1/25

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is going to be really close to
there-- Then y is negative 2.

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So it's going to be
like right over there,

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not quite at the y-axis.

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

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But pretty close.

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So that's right over there.

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That is 1 over 25, comma
negative 2 right over there.

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Then, when x is one
fifth, which is slightly

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further to the right, one
fifth y is negative 1.

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So right over there.

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So this is one
fifth, negative 1.

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Then when x is 1, y is 0.

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So 1 might be right there.

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So this is the point 1,0.

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And then when x is 5, y is 1.

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When x is 5, I covered it
over here, when this is five,

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y is 1.

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So that's the point 5,1.

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And then finally,
when x is 25, y is 2.

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So this is 25,2.

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And then I can
graph the function.

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And I'll do it-- let me do it
in a color-- I'll use this pink.

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So as x gets super, super,
super, super small, y goes

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to negative infinity.

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It gets really small-- to
get x's or as x becomes--

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if you say what power do
you have to raise 5 to

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to get 0.0001?

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It has to be very, very,
very negative power.

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So y is going to be very
negative as we approach 0.

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And then it kind of
moves up like that.

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And then starts to kind of
curve to the right like that.

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And this thing
right over here, is

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going to keep going down at
a steeper and steeper rate.

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And it's never going
to quite touch.

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the y-axis.

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It's going to get closer
and closer to the y-axis.

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But it's never going
to be quite touch it.