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Video transcript

- [Narrator] Right over here, we've defined y as a function of x, where y is equal to the natural log of x minus three. 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 we'll talk about, and 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, this thing right over here, must be positive, or, another way we could think about it, it's only going to be defined for any x, such that, x minus three is strictly positive. Not greater than or equal to zero, greater than zero. We haven't defined what the natural log of zero is, so x minus three has to be greater than zero in order to give a positive, in order to take the natural log of a positive something, or if we add three to both sides, we get x is greater than three. So, it defined, defined four, x greater than three. You could do this as the domain, the set of all real numbers that are greater than three. Now, that out of the way, let's actually try to plot natural log of x minus three. So, let me put some of this graph paper right over here, and the first thing I wanna think about is, well, let's just try to plot some interesting points here, and the most obvious one is what makes this natural log, what makes this entire function equal to zero? 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 three going to be equal to zero? 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 three is the same thing as e to the zero, and, of course, if you raise e to whatever exponent you need to get you to x minus three, that's just going to get you to x minus three, and if you raise e to the zero, well, anything to the zeroth power, except possibly zero, that one's under contention, or maybe not defined, e to the zero is equal to one. This is just another way of saying, "Hey, look, if I wanna know what exponent "do I need to raise e to to get to zero," we know e to the zeroth power is equal to one, so x minus three is equal to one. So, if I'm taking the natural log of one, it'll be zero. So, x minus three is equal to one, add three to both sides, you get x equals four. So, we know that the .4, zero is on this graph, so let me graph that. One, two, three, and four. So, that right over there is the point. X is four, and y is zero. Four minus one, four minus three is one, natural log of one is zero. We also know that this is only defined for x being greater than three, so let's just put a little dotted line right over here at x equals three. And we know that our function isn't even defined, for x equals three, and in any value, to the left of it. Well, let's think about what happens as we approach x equals three from the right-hand side, and to do that, I'll make a little table here. So, let me make a little table here, and put some x values here, and then let's just think about what our corresponding y value is. So, we could put in, so we already know that we get four, zero. Let's try out 3.1, 3.01, and 3.001 and see what you get. And you can imagine, from each of these, you're gonna subtract three, so then the input into the natural log function is gonna be .1, .01, .001, and so you're gonna 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. Let's get our calculator out, and let me go to the main screen. And so, let's take the natural log of three, of 3.1 minus three. 3.1 minus three. We get negative 2.3, and I'll just round to the tenth. So, this right over here is negative 2.3. If we raise, if we take the natural log, natural log of 3.01, 3.01 minus three, we get to negative 4.6, once again, just rounding, negative 4.6, and if we take the natural log, the natural log of 3.001, zero, one, minus three, we get to negative 6.9. 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., let's do one, two, three, four, five, zeros, followed by one, minus three, and we get a fairly more negative value right over here. So, as you see, as we're getting closer and closer to three, we're getting more, and more, and more negative values. Let me just plot this right over here, so this is negative one, this is negative two, this is negative three, this is negative four. 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, so it's way down here. So, our graph is gonna look something like, our graph is gonna look something like, and my best attempt is to draw it freehand, is gonna look something, something like, something like that. So, do we have a vertical asymptote? Absolutely. As we approach three from values larger than three, 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. We have a vertical asymptote at, asymptote, at x equals three.