>> This is Graphing
Logarithmic Functions.
And before we start
any graphing,
we need to know what are the
actual X values we can plug
in here, and that means we're
looking for the domain first.
So if we go back to the
domain, we're supposed
to take the inside and
make it greater than 0.
So we take the inside, we
make it greater than 0.
So that is our domain.
X must be greater than 0, which
means we can't start with 0
but we want something
close to it, so 0.1.
Just add 0.1 to it
if we're going
in the greater than direction.
And then just keep
counting from there.
And that should be good enough.
I have extra spots
if we want more.
And then to plug it in, we
have to use a change of base
because there is no log
base 2 on our calculator.
So, log of X over log of 2.
And then now we can
plug in the 0.1
but we will always be
dividing by log of 2.
And so we're going to plug
in 0.1 into our calculator.
So we go, log 0.1
divided by log of 2.
Close the parenthesis, hit
enter, and there it is.
Negative 3.32, and
that's all we need.
So, negative 3.32.
And they're [inaudible]
a log of 1 is 0.
So now we go on to log of 2.
Clear. And it's going to be
log of 2 divided by log of 2.
Since it's the same thing
we know it's going to be 1.
And now we just need one more.
Clear this.
So log of 3, close it,
divided by log of 2,
close the parenthesis, 1.58.
And there it is.
If you wanted to do a 4,
it would give us
the same basic idea.
And clear this.
Log of 4 divided by log of 2.
And we get 2.
So you see, it just keeps
getting bigger and bigger
and it's going to keep growing.
So when we graph it,
we're supposed to take
and label our vertical
asymptote at 0.
So let me grab some right
here It's a little bit thick.
So at 0, there is a
vertical asymptote.
So we know never to
cross this graph.
Remember, it can never equal 0.
So now when we plot
this, it's 0.1,
which is very tiny here,
very close in there.
Oops. Very close, right there.
And then we have 1, 0 and
then 2, 1, and then 3, 1.58,
and you could see how
it's growing that way.
And then 4 is 2, and
there's our graph.
So make sure you
never cross down here.
It just gets closer, and then
the other side goes off quickly
that way.
All right.
Let's do another one.
So here, we have X plus
2 must be greater than 0,
and solving it means X is
greater than negative 2.
So that's our domain.
So, remember, we can't
plug in negative 2.
So what we have to plug
in is something that's
a little bit bigger
which would be negative 1.9.
So we're going that way.
You can't do negative
2.1, that's not bigger.
So, negative 1.9
and then negative 1,
and then 0 and then 1.
That should be good enough.
So let's plug them in.
So we have negative log
of negative 1.9 plus 2.
Now, and again, you got
to divide it by log of 2.
[ Pause ]
And there we go, 3.32.
And then the negative 1 will
give us 1 so that again is 0.
Plug in 0.
We will get 2, and log
of 2 over log of 2 is 1.
So you get negative 1.
And you can enter all these in
your calculator if you want.
Just do the last one
which is 1 to clear it.
So, negative, negative log and
then we have to do 1 plus 2,
close it, divided
by log of 2, enter.
And we get negative 1.58.
All right.
So let's draw our
vertical asymptote
and it's at negative 2.
Right there.
And we can label it as
X equals negative 2.
We can label that
one as X equals 0,
those are our vertical lines.
And then now we get to graph.
So we're going to go negative
1.9, that's really close
to the asymptote and then
it's 3.32 right there.
And then it's negative
1, 0, 0, negative 1,
and then 1, negative 1.58.
Do you notice now that
these are just going
in the opposite direction?
So, we are going
up and then down.
Like that.
It's because of the negative.
Remember the negative
flips it across the X-axis,
so it takes it from here
over here and so on.
So, our negative
values become positive
and our positive become negative
because we have an
opposite here.
So instead of having a negative
3.32, we have a positive 3.32.
And instead of a positive
1, we have a negative 1.
Instead of a positive 1.58, we
have a negative 1.58, and so on.
So, if we would have
kept going, keep going,
at 2 it would be negative 2.
OK? But that's it.
This is all you have to do.
You need a vertical
asymptote and it's the number
from our domain and then
you just pick some values
from that domain.
So there's our steps.
Find the domain and find
the vertical asymptote
and label it X equals
0 or X equals 2.
It's based on our domain.
And then plot some points.
Remember you want to
pick one that's close,
and then just keep counting out.
OK. Let's try one more,
this one is a base 3
and then I'll let
you try that one.
All right.
So, with a base 3, we
got to find our domain.
So you have X plus
1 is greater than 0,
so X is greater than negative 1.
OK. So we can't start
with negative 1.
So, what's one that's
very close?
Negative 0.9.
You're basically
just adding 0.1.
So if you don't know where
to start, take negative 1,
1 negative, and then add 0.1.
That gets you really
close to it.
OK? So our vertical
asymptote is that negative 1.
Let's draw it.
Let's put that in red so we
know where it's coming from.
It's right there.
So that is our line that we
cannot cross, right there.
And we're going to
plug in some points.
So, negative 0.9 and then
the next one off is 0
and then 1 and then 2.
And this time we have
to divide by log 3.
So the formula we're
going to use is log
of X plus 1 over log of 3.
So I want you to plug those
into your calculator and see
if you can get the answer.
So remember make-- I'll
give you the first one.
Make it exactly how it looks.
It's log of negative 0.9, right?
Plus 1, parenthesis and
then divided by log of 3
because that's our new
base, so always dividing
by that base, and there it is.
Negative 2.095.
Now you got to plug
in 0 and 1 and 2.
Pause, give them a shot.
So you should have these points
and a rough graph
that looks like that.
Now try the next one.
Hit pause and see if you can
do the whole thing by yourself.
Now, it does have a little
trick here as 1 minus,
so to give you a helping hand,
this is the formula you're
going to be plugging into.
You just change the base
on the logarithm like that.
And that's what you
should have for your graph.
And just one more thing,
and that's domain and range.
And as you can see with the
domain, we already found it
from the very beginning and it's
X is greater than negative 1.
So you want to write
negative 1 comma infinity.
Or you can say X is
greater than negative 1.
And then over here, it's
X is greater than 2.
So it would be from
2 to infinity
where X is greater than 2.
But you must use parenthesis
because they will never
actually hit that value.
Remember, it can never cross the
value that gives it a 0 inside.
And then the range is nice,
negative infinity to infinity
and then negative
infinity to infinity.
As you keep plugging in points,
you will always get much more
and more and more negatives,
as you keep getting closer
and closer to the asymptote.
And as you keep going farther
and farther out with your X's,
you're going to get
more and more Y's
and it will just keep
going on forever.
Same with this one
to go on forever
and it will go on forever up.
So we're going to have as
many negatives as we want
and as many positives we
want, all real numbers.
Thank you.
