Graphing Logarithmic Functions

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

And just to remind us
what this is saying,

this is saying that y
is equal to the power

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

Or if I were to write
this logarithmic equation

as an exponential
equation, 5 is my base,

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

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

So another way of writing
this equation would be 5

to the y'th power is
going to be equal to x.

These are the same thing.

Here, we have y as
a function of x.

Here, we have x as
a function of y.

But they're really saying
the exact same thing,

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

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

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

We'll have to raise it to y.

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

I get x.

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

that we can use to
plot some points,

and then we can
connect the dots to see

what this curve looks like.

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

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

some nice round answers, some
nice fairly simple numbers

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

have to get the calculator.

And so in general,
you want to pick

x values where the power
that you have to raise 5

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

Or another way to
think about it,

you could just think about
the different y values

that you want to raise
5 to the power of,

and then you could
get your x values.

So we could actually
think about this one

to come up with our
actual x values.

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

the independent variable is x,
and the dependent variable is

y.

We might just look at
this one to pick some nice

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

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

just so we get nice clean x's.

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

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

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

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

I'll do one more, and then 2.

So once again, this is
a little nontraditional,

where I'm filling in the
dependent variable first.

But the way that
we've written it over

here, it's actually given
the dependent variable,

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

to be for this
logarithmic function.

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

What x gives me--
what does x have

to be for y to be
equal to negative 2?

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

So 5 to the negative
2 is 1 over 25.

So we get 1 over 25.

If we go back to
this earlier one,

if we say log base
5 of 1 over 25,

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

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

Or you could say 5
to the negative 2

is equal to 1 over 25.

These are saying the
exact same thing.

Now let's do another one.

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

I get one fifth.

So if we go to this
original one over there,

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

Want to be careful.

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

in order to get one fifth.

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

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

I get one.

And so this relationship--
This is the same thing

as saying log base 5 of 1.

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

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

Let's do the next two.

What happens when I raise
5 to the first power?

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

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

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

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

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

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

to get to 25?

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

So I took the inverse of
the logarithmic function.

I wrote it as an
exponential function.

I switched the dependent
and independent variables,

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

y's.

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

I could have just picked
random numbers over here,

but then I would have probably
gotten less clean numbers

over here.

I would have had to
use a calculator.

The only reason why
I did it this way,

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

So let's actually graph it.

Let's actually graph
this thing over here.

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

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

So let's graph it.

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

Draw it like that.

That is my x-axis.

And then the y's start at 0.

Then, you get to
positive 1, positive 2.

And then you have negative 1.

And you have negative 2.

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

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

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

is a logarithmic
function defined

for an x that is not positive?

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

No.

You could raise five to an
infinitely negative power

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

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

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

So x cannot be 0.

And there's no power
then you could raise 5

to get another negative number.

So x can also not be
a negative number.

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

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

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

Let me write that down.

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

So we're only going
to be able to graph

this function in
the positive x-axis.

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

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

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

And then let's plot these.

So the first one is in blue.

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

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

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

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

not quite at the y-axis.

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

But pretty close.

So that's right over there.

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

Then, when x is one
fifth, which is slightly

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

So right over there.

So this is one
fifth, negative 1.

Then when x is 1, y is 0.

So 1 might be right there.

So this is the point 1,0.

And then when x is 5, y is 1.

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

y is 1.

So that's the point 5,1.

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

So this is 25,2.

And then I can
graph the function.

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

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

to negative infinity.

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

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

to get 0.0001?

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

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

And then it kind of
moves up like that.

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

And this thing
right over here, is

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

And it's never going
to quite touch.

the y-axis.

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

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