Vertical asymptote of natural log | Limits | Differential Calculus | Khan Academy

Right over here, we've defined y as a function of x, where y is equal to the natural log of x - 3. What I encourage you to do right now is to pause this video and think about for what x values this function is actually defined. Or another way of thinking about it is, what is the domain of this function? Then, try to plot this function on your own, on maybe some scratch paper that you might have in front of you, and then we'll talk about it. Also, think about whether this has any vertical asymptotes, and if so, where.

So, I'm assuming you've given it a go at it. First, let's think about where this function is defined. 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.

Another way we can think about it is that it's only going to be defined for any x such that x - 3 is strictly positive, not greater than or equal to zero. We can't define what the natural log of 0 is, so x - 3 has to be greater than zero in order to take the natural log of a positive number. If we add 3 to both sides, we get x is greater than 3. So, defined for x greater than 3. You can view this as the domain: the set of all real numbers that are greater than 3.

Now that that's out of the way, let's actually try to plot natural log of (x - 3). Let me put some graph paper right over here. The first thing I want to think about is, well, let's just try to plot some interesting points here.

The most obvious one is what makes this natural log equal to zero? When are we going to intersect the x-axis? So, let's just think about that for a little bit. When is the natural log of (x - 3) going to be equal to 0?

Well, one way to think about this is to view both sides as exponents and raise e to both of these powers. You could say that e to the natural log of (x - 3) is the same thing as e to the 0. Of course, if you raise e to whatever exponent you need to get x - 3, that's just going to give you x - 3. If you raise e to the zero, well, anything to the zero power, except possibly zero—which is 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 I need to raise e to get to zero, we know e to the zeroth power is equal to 1." So, x - 3 is equal to 1. If I'm taking the natural log of 1, it'll be zero. Therefore, x - 3 equals 1. Adding three to both sides gives us x = 4. So we know that the point (4, 0) is on this graph.

Let me graph that: 1, 2, 3, and 4. So, that right over there is the point (4, 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 = 3. We know that our function isn't even defined for x = 3 and 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. To do that, I'll make a little table here. Let me make a table and put some x values here, and then let's just think about what our corresponding y value is.

We could try so we already know that we get (4, 0). Let's try out 3.1, 3.01, and 3.1, and see what you get. You can imagine from each of these you're going to subtract three, so the input into the natural log function is going to be 0.1, 0.01, and 0.1. You will have more and more negative values that you have to raise e to in order to get to those values.

But to just verify that, let's actually get our calculator out. Let's go to the main screen and take the natural log of (3.1 - 3). We get approximately -2.3. I'll just round to the tenth. So this right over here is -2.3.

If we take the natural log of (3.01 - 3), we get about -4.6, negative once again, just rounding. If we take the natural log of (3.001 - 3) we get approximately -6.9. Just for fun, let's do one that's way more dramatic.

So, let's take the natural log of (3.0001 - 3) and we get a much more negative value right over here. As you see, as we're getting closer and closer to 3, we're getting more and more negative values.

So, let me just plot this right over here. So this is 1; this is 2; this is 3, and let's say this is -4. So, when x is equal to 3.1—which is right about there—we're at -2.3. When x is 3.01, we get to -4.6, which is way down here.

So, our graph is going to look something like this. 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; our function values are quickly approaching negative infinity.

So, we clearly have a vertical asymptote at x = 3.