 We can use exponential functions to model many different phenomena, such as decay of radioactive substances, metabolism of drugs, spread of diseases, growth of death, and so on. Now there are actually two ways we could define an exponential function. First, let a and b be positive constants with b not equal to 1. Then f of x equals a b to power x is an exponential function. But actually it's convenient to use e as our base, and so with a little bit of algebra we can transform this b to power x into e to power kx, and we get a different definition, let a and k be non-zero constants with a greater than 0, then f of x equals a e to power kx is an exponential function. One place these exponential functions show up is in something called half-life. The half-life of a substance is the time it takes for half of it to be broken down. For example, a drug in the bloodstream has a half-life of 6 hours. If the initial concentration in a patient's bloodstream is 24 micrograms per liter, what will the concentration be in 8 hours? So we want to find an exponential model here, so let c of t be the concentration after t hours, then c of t is equal to a e to power kt. And to write the exponential model, we need to find a and k. Now we have c of t is the concentration after t hours, and we're given this initial concentration of 24 micrograms per liter. And since this is the initial value, this is really the concentration after 0 hours, and we know that's equal to 24. But wait, remember that equals means replaceable, and we replaced t with 0. And so any place we see a t, we can replace it with 0. Well, how about here? So c of 0 is the concentration after 0 hours equals means replaceable. There's a t in our formula for c of t, so we'll replace it equals means replaceable. So c of 0 is our concentration after 0 hours, which is 24. We'll do a little algebra, k times 0 is 0, and e to power 0 is 1. So the right-hand side simplifies, and we find that a is equal to 24. Now the half-life of 6 hours means that after 6 hours, the amount is halved. And so c of 6, well that's a concentration after 6 hours, and if it was 24 micrograms, then its concentration will be 12 micrograms, and equals means replaceable. c of 6, well according to our formula, and the fact that we know what a is, well that's going to be 24e to power k times 6. We know c of 6, and we can solve for k. And an important idea here is don't round until you reach the final answer. So we can leave this in the form log 12 divided by 24 divided by 6, until we enter it into a calculator to get our final answer, which we have to round in order to be able to write it down. And finally we want to know the concentration after 8 hours. So again, c of t is the concentration after t hours equals means replaceable. We want this time to be 8 hours, so we'll replace t with 8, everywhere we see it, here and here, and calculate.