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# Vertical asymptote of natural log

Video transcript

Right over here, we've
defined y as a function of x, where y is equal to
the natural log of x minus 3. What I encourage you to do
right now is to pause this video and think about for
what x values is this function actually defined. Or another way of
thinking about it, what is the domain
of this function? And then, try to plot
this function on your own on maybe some scratch paper that
you might have in front of you. And then, also think
about, does this have any vertical asymptotes? And if so, where? So I'm assuming you've
given a go at it. So first, let's think about
where is this function defined? So the important
realization is this is only defined if you're
taking the natural log of a positive value. So this thing right over
here must be positive. Or another way we
could think about-- it's only going to be defined
for any x such that x minus 3 is strictly positive. Not greater than or equal
to 0, greater than 0. We haven't defined what
the natural log of 0 is. So x minus 3 has to be
greater than 0 in order to take the natural log
of a positive something. Or if we add 3 to both sides,
we get x is greater than 3. So defined for x
greater than three. You could view this
as the domain-- the set of all real numbers
that are greater than 3. Now, with that out the
way, let's actually try to plot natural
log of x minus 3. So let me put some graph
paper right over here. And then the first thing
I want to think about is, well, let's just try to plot
some interesting points here. And the most obvious
one is-- What makes this entire
function equal to 0? When are we going to
intersect the x-axis? So let's just think about
that for a little bit. So when is the natural log of x
minus 3 going to be equal to 0? Well, one way to
think about this is to view these
both as exponents and raise e to both
of these powers. So you could say that e to
the natural log of x minus 3 is the same thing as e to the 0. And of course, if you
raise e to whatever exponent you need to
get you to x minus 3, that's just going to
get you to x minus 3. And if you raise
e to the 0, well, anything to the 0-th power,
except possibly 0, that one's under contention or maybe
not defined-- e to the 0 is equal to 1. This is just another way
of saying, hey, look, if I want to know what
exponent do I need to raise e to to get to 0, we know e to
the 0-th power is equal to 1. So x minus 3 is equal to 1. So if I'm doing the natural
log of 1, it'll be 0. So x minus 3 is equal to 1. Add 3 to both sides. You get x equals 4. So we know that the point
4 comma 0 is on this graph. So let me graph that. 1, 2, 3, and 4. So that right over
there is the point. x is 4, and y is 0. 4 minus 3 is 1. Natural log of 1 is 0. We also know that
this is only defined for x being greater than 3. So let's just put a little
dotted line right over here at x equals 3. And we know that our
function isn't even defined for x equals 3 and at any
value to the left of it. But let's think
about what happens as we approach x equals 3
from the right-hand side. And to do that, I'll
make a little table here. So let me make a table here
and put some x values here. And then, let's just think
about what our corresponding y value is. So we already know
that we get 4, 0. Let's try out 3.1, 3.01, and
3.001, and see what you get. And you could imagine
from each of these you're going to subtract 3. So then, the input into
the natural log function is going to be 0.1, 0.01, 0.001. And so you're going to
have more and more negative exponents or powers
that you have to raise e to to get to those values. But to just verify
that, let's actually get our calculator out. And let me go to
the main screen. And so let's take the
natural log of 3.1 minus 3. We get negative 2.3. And I'll just
round to the tenth. So this right over
here is negative 2.3. If we take the natural
log of 3.01 minus 3, we get to negative 4.6. Once again, just rounding. Negative 4.6. And if we take the natural
log of 3.001 minus 3, we get to negative 6.9. And just for fun, let's do
one that's way more dramatic. So let's take the
natural log of 3 point-- let's do 1, 2, 3, 4, 5
zeroes followed by 1 minus 3, and we get a fairly more
negative value right over here. So as you see, as we're
getting closer and closer to 3, we're getting more and more
and more negative values. Let me just plot
this right over here. So this is negative 1. This is negative 2. This is negative 3. This is negative 4. So when x is equal to 3.1--
which is right about there-- we're at negative 2.3--
which is right around there. When x is 3.01-- which is really
hard to see right over here-- we get to negative 4.6--
which is way down here. So our graph is going to
look something like-- my best attempt to draw it freehand--
is going to look something like that. So do we have a
vertical asymptote? Absolutely. As we approach 3 from
values larger than 3-- from the right-hand side-- our
function is plummeting down. It's unbounded. It's going down. Our value of our
function is quickly approaching negative infinity. So we clearly have a vertical
asymptote at x equals 3.