>> Here's how we build on this, this general idea. Here's our, here's the model that we're building on, and let's say that in a city in 1990, the population of the city was two million, and in the year 2000, three million. We'd like to know the approximate population in the year 2010. Now to, to start the process, we, we think about our model this way. Now, A is the starting population. Y is going to be the ending population, and X is going to be time. Alright, and then B is going to be some constant that tells us about how fast we're growing. Graphically, it's easy to see graphically. The kind of shape of that graphical curve. And so, we need to find the value of B early on in this situation, and we find that by performing some kind of an experiment or by using the given information. Alright. Here's how we would use the given information. Now, you know that these numbers can correspond to ordered pairs, and we have to start with a, when we have time-related problems, we start with a time basis, and the basis here is 1990. So we would let 0 stand for this time, and at that point in time, the population is two million, and for the year 2000, since it's 10 years from 1990, ten years have elapsed here. So ten corresponds with the number 3. So we have a couple of ordered pairs. But here's how we can jump right into the model. For the year, in the year 2000, you see, we have start, the start population and the end population for the year 2000. Here's how the model works. The starting population was in 1990, and it was two million people. The ending population was in the year 2000, and it was three million people. Alright. Now, the time is ten years for this period of time. That is, from 1990 to the year 2000. Now, this with this, we're going in, and we're trying to solve this for B. We're trying to get at that sort of, when we were talking about this in variation, we were looking for the constant of proportionality, and here, this is kind of a, a constant of exponentiality [phonetic], you see, is what we're looking for here, and we're using the information from our experiment or the, the information that's, that's given here from a historical standpoint. Alright. We just need to solve this equation. I'm dividing on both sides by 2 to isolate E to this exponent. Now, I need to jump into the exponential world by applying natural log on both sides. Now, we have the natural log of one and a half is equal to 10B. Dividing on both sides by 10, I find this fraction to be equal to B. Now, we could at this point approximate this fraction, but we're going to use this back in our model, and if we're going to use this, then it's, it's valuable for us not to approximate because if you use an approximation, and then operate on it, perform operations on it, it turns out that you can multiply the error because an approximation contains an error, however small. So, let's use the exact value here for B. So we would come up here, we would take our model, and now that we know the value of B, we would put it in. Now, notice I've, I've, I'm using T as the variable here instead of X because that's the nature of this particular problem. Now, with this, we can answer any number of questions involving time or population. We can say, OK, I'd like to know the time when the population will reach, say five million, or something like that. And so I can put in 5 here, and solve for T, or I can take a period of time, and put in here, and solve for Y. That is, the population after a certain period of time. Now, the nature of our, our problem is that we're looking for the population in the year 2010. So let's see. What would T become? Well, we started as a basis year, 1990, and so you, you calculate a period of time from 1990. So from 1990 to 2010 is 20 years. So we use 20 for T, and away we go, you see, and just, this is just a calculation, a calculator push at this point. It looks a little complicated, but it's not really a big deal. We find that Y, then is approximately 4.5 million, and that means that the population of this city is predicted to be approximately 4.5 million people in the year 2010.