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So this is going to be an example of an exponential function, and I'll show you why. So we have a population of bacteria doubles every 30 minutes. So if we think about this, we're starting with one bacterium, which is just singular bacteria. So a T equals zero. We have one. Let's say that pft is the number after 30 minutes. So what I'm really doing here is I'm gonna make a table of values. So we have tea and then one of pft. So at zero, we have one, and it doubles, Okay, every 30 minutes. So after 30 minutes, there should be two. And then after another 30 minutes, So after 60 total minutes, there should be four. And then after 90 there should be eight etcetera. Okay, so why did I do that? Well, that actually helps me figure out what pft is because notice that every 30 minutes we're going to double, so that means p of tea should be. Well, what? I started with one and then times two. How many times? Um, I'm gonna multiplied by two. Well, I'm gonna multiply by two once for every 30 minutes, so I'm going to put t divided by 30 because when t is 30 I'm going to get to to the first two and t is 60. I'll have tea to the second, which is four. When he is 90 I'll have tea cubed, which is eight etcetera. So pft is just to to the t divided by 30. Okay, so this is a exponential function. Now I want to know how long it will take for the bacteria to reach a population of 100. So in other words, I want to find t for which pft equals 100. So, in other words, I 100 to be to to the t over 30 and I need to solve for T. So this is going to illustrate the inverse relationship between exponential zand logarithms I'm trying to solve for t. I need toe isolate teeth, but T is up in the exploding oven exponential function. But what I can do is apply log base to, and I'm using log based too, because the base of the exponential too, to both sides. So, in other words, this equation is equivalent to saying log based two of 100 is equal to log base to of to to the t over 30. But Log based too of two to the T over 30 is just going to be t over 30 because the log based to in the exponential based to cancel out. And so then I just multiply both sides by 30. So I get the tea is 30 times log base to of 100. Okay, well, that's just a number. Let me tell you approximately what it is. It's going to be approximately 199 0.3 minutes, and there's our answer. So after well, about 200 minutes, the population of this bacteria should be at 100.

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