Exponential model word problem: medication dissolve | High School Math | Khan Academy

Carlos has taken an initial dose of a prescription medication. The relationship between the elapsed time T, in hours, since he took the first dose, and the amount of medication m, in milligrams, in his bloodstream is modeled by the following function:

In how many hours will Carlos have 1 milligram of medication remaining in his bloodstream? So, M of T is equal to... So we need to essentially solve for M of T, which is equal to 1 milligram, because M of T outputs whatever value it outputs, and it's going to be in milligrams. So let's just solve that.

M of T is defined; its model is an exponential function: 20 * e^(0.8T) = 1. So let's see if we can divide both sides by 20. Then we will get e^(0.8T) = 1/20, which we could write as 0.05. I have a feeling we’re going to have to deal with decimals here regardless.

So how do we solve this? Well, one way to think about it is if we took the natural log of both sides. Just as a reminder, the natural log is the logarithm base e. Let me rewrite this a little bit differently. So this says 0.05. Now I’m going to take the natural log of both sides.

So, Ln, Ln… The natural log says what power do I have to raise e to, to get to e^(0.8T)? Well, I’ve got to raise e to the negative 0.8T power. So that’s why the left-hand side simplified to this.

And that’s going to be equal to the natural log… Actually, I'll just leave it in those terms: the natural log of 0.05. Now we can divide both sides by 0.8 to solve for T. So let's do that.

We divide by 0.8, and so T is going to be equal to all this business on the left-hand side. Now we just have a T, and on the right-hand side, we have all this business, which I think a calculator will be valuable for.

Let me get a calculator out, clear it, and let’s start with 0.05. Let’s take the natural log—that’s that button right over there. The natural log, we get that value. Now we want to divide it by -0.8.

So, divided by -0.8, so let's see… They want us to round to the nearest hundredth. So it will take approximately 3.74 hours for his dosage to go down to 1 milligram.

It actually started at 20 milligrams when T equals 0. After 3.74 hours, he’s down to 1 milligram in his bloodstream. I guess his body has metabolized the rest of it in some way.