 Hello, and welcome to a screencast about exponential applications. Alright, this problem is about a lab experiment, and we need to grow a culture of 150,000 bacteria, which are known to double in number every four hours. If we begin with 1,000 bacteria, how many hours will it take to reach the desired number of bacteria? Alright, so I like to make a table with these types of problems, just so I can kind of see what's going on. So let me read these a little bit more carefully again. So if we begin with 1,000 bacteria, okay, so what does that mean in terms of my T time and my P population? Well, beginning means time is zero, population is 1,000 bacteria. Okay, we're also given the fact that it's known to double in number every four hours. Okay, so what does that mean in terms of T and P? Well, four hours have passed in our population, and our number of bacteria has doubled. So double of 1,000, that gives us 2,000. Okay, if we want to keep going with our table, we'll just do one more and just make sure you guys get the hang of it. Another four hours will have passed, so that'll be eight hours, and then we'll double our 2,000, that would give us 4,000. Okay. Alright, but now we want to go ahead and go to an equation. So our equation for these exponential growth and decay is P of T equals P naught, P sub zero, E to the K times T. Okay, and what these things stand for? So P of T is your population at some point in time. P sub zero here, that's your initial population, which we know for this particular problem. The K up here in our exponent is our growth constant, or decay constant, depending on the problem. And then the T up here in our exponent is time. Okay, so for this particular problem, time is measured in hours, and you will want to make sure those variables match up. Okay, and you notice I skipped over E, and that's because E is actually a number. It's a constant in math, you know, so you can use your exponential button there. Okay, so now that we know how to use this equation, let's go ahead and start plugging stuff in. So our population after a certain amount of time, well you can use either 2,000 or 4,000, whatever one you want to use. I'm going to use the 2,000 though, since that's our first one we made up. Our initial population is 1,000. We've got our E, K we don't know, and our time is four hours. Okay, so now we've just got to use our methods that we've been learning in class to be able to solve for the variable up here in the exponent. So the first thing we want to do is get rid of our coefficient. So let's divide both sides by 1,000 to get that E by itself, and that gives us a 2, which makes sense because we're doubling E to the K times 4. Now how do we undo the E function? Well, it's an exponential base E, so that means we need the logarithm base E, which is our natural log. Okay, so that'll wipe that stuff out. So we end up with the natural log of 2 equals K times 4, divide both sides by 4, crunch that in the calculator, and we get a K value of about 0.1733. Okay, and that's approximate. If you want to use the exact value, go for it, but approximate does smell a little bit easier. All right, so now what are we trying to do with this problem? If we go back and read it, we need to grow a culture of 150,000 bacteria. So that's a population value after a certain amount of time, which is what we're trying to figure out. So we're going to have 150,000 equals our initial value, which is 1,000, E to the K, which we just figured out 0.1733 times our time, which is T. And we're going to solve it pretty much the exact same way we just solved the last one. Okay, so again, go ahead and divide both sides by 1,000, get lots of good practice with these. Get rid of our coefficient, 150,000 divided by 1,000 gives us 150 equals E to the 0.1733T. Again we need to undo that E function with our natural log function, so those will cancel. So we'll end up with a natural log of 150 is going to equal 0.1733T. Divide both sides by 0.1733 to get the T by itself. Crunch us on the calculator and we get about, let's see, 29 hours. And it's not exactly 29, it's about 29, but it's somewhere close in there. Alright, thank you for watching. And here are our credits, so thank you to Grand Valley for creating this wonderful YouTube channel.