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Studying for a test? Prepare with these 31 lessons on Exponential & logarithmic functions.
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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.