Exponential model word problem: medication dissolve

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 all right and how many hours will Carlos have one milligram of medication remaining in his bloodstream so M of what T is equal to so we need essentially 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 its model is an exponential function 20 times e to the negative 0.8 T is equal to 1 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 1 over 21 over 20 which we could write as 0.05 0.05 I'm feeling we 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 if we took what happens if we took the natural log of both sides and just remember the 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 that is 0.05 so I'm going to 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 zero point eight T well I've got to raise e to the this this simplifies to negative zero point eight T once again natural log this thing let me clarify Ln of e to the negative zero point eight T this is equivalent to if I were to write log base e of e to the negative 0.8 t what power would have to raise e to to get to e to the negative zero point eighty we 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 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 divided 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 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 by negative 0.8 so divided by divided by 0.8 negative so we're going to divide by 0.8 negative is equal to let's see they want us to round to the nearest hundredth so three point seven four so it'll take three point seven four seven four hours for his dosage to go down to one milligram where it actually started it 20 milligrams when T equals zero it's 20 after three point seven four hours he's at down in his bloodstream to 1 milligram I guess his body has metabolized the rest of it in some way