Exponential model word problem: medication dissolve

- [Voiceover] Carlos has taken an initial dose of a prescription medication. The relationship between the time, between the elapsed time t, in hours, since he took the first dose and the amount of medication, M of t, in milligrams, in his bloodstream is modeled by the following function. Alright, in how many hours will Carlos have 1 milligram of medication remaining in his bloodstream? So M of what t is equal to, so we need essentially to solve for M of t is equal to 1 milligram. Because M of t outputs, whatever value it outputs is going to be in milligram. So let's just solve that. So M of t is, they give us a definition, it's model is an exponential function, 20 times e to the negative 0.8 t is equal to one. So let's see, we can divide both sides by 20 and so we will get e to the negative 0.8 t is equal to one over 20, one over 20. Which we could write as 0.05, 0.05. I have a feeling we're gonna have to deal with decimals here regardless. And so how do we, how do we solve this? Well one way to think about it, one way to think about it if we took, what happens if we took the natural log of both sides? And just a remember, a reminder, the natural log is the logarithm base e. So actually let me write this, let me write this a little bit differently. So this is zero and that is 0.05. So I'm gonna take the natural log of both sides, so ln, ln. So the natural log, this says, what power do I have to raise e to, to get to e to the negative 0.8 t? Well I've got to raise e to the, this simplifies to negative 0.8 t. Once again natural log this thing... let me clarify ln of e to the negative 0.8 t. This is equivalent to if I were to write log base e of e to the negative 0.8 t. What power do I have to raise e to, to get to e to the negative 0.8 t? Well I have to raise it to the negative 0.8 t 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, the natural log of 0.05 all of that and now we can divide both sides by negative 0.8 to solve for t. So let's do that. So we divide by negative 0.8, divide by negative 0.8 and so t is going to be equal to all of this business. On the left-hand side now we just have a t and on the right-hand side we have all of this business which I think a calculator will be valuable for. So let me get a calculator out. Clear it out 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 and we want to divide it by negative .8. So divided by, divided by .8 negative. So we're gonna divide by .8 negative is equal to, let's see they wanted us to round to the nearest hundredth so 3.74, so it'll take 3.74, seven four hours for his dosage to go down to one milligram where it actually started at 20 milligrams. When t equals zero it's 20 after 3.74 hours he's down in his bloodstream to one milligram. I guess his body has metabolized the rest of it in some way.