 In a perfect world, we'd always get information about an exponential function at t equal to zero. We'd always begin at the beginning. We'd know exactly the amount we started with. We'd catch pandemics on the day they started. We don't live in that world. But we can use transformations to bring us there. For example, a disease begins spreading. The number of persons infected t days after the start is recorded as follows. Let's find an exponential model, then determine how many persons are infected after 60 days, and then when will the number infected reach 100,000? So if we assume an exponential model of the form n of t equals a e to power k t, our given information tells us the following. At t equals 25, n of 25 is equal to 8, and at t equals 40, n of 40 is equal to 11. And so we know that n of 25 is a e to power 25k, and n of 40 is a e to power 40k. And since we know n of 25 and n of 40, we can replace the left-hand side. And so we need to solve the system of the two equations. And we could do this, but we don't want to. Now you might notice that it's easier to solve an exponential equation when the exponent is 0, and we can make our exponent 0 if we translate. Remember, f of x minus h represents a horizontal translation of the graph of y equals f of x. And so we'll let our function be n of t equals a e to power k t minus 25, where we use t minus 25 because we start at t equal to 25. So if n of t is a e to power k times t minus 25, we know what happens at 25. And we know that n of 25 is equal to 8, so that gives us a. And we also know that n of 40 is equal to 11, and since we already know the value of a, we can solve for k. And since we know the value of a and k, we can now find at t equal to 60, we have. And here, because n of t represents thousands of cases, it doesn't make sense to carry that fourth decimal place. And so at t equals 60, there's approximately 16.819,000 persons infected. To find when the number infected will reach 100,000, well, we want 100,000 people infected. Well, actually, we don't. We're asked to find when 100,000 people will be infected, so we know our function, and we have that 100,000 people are infected, and we'll solve for t. And again, it's important for purposes of accuracy, not to round until we get to our final answer. And since t is the number of days, we will round this to the nearest whole number. Now, when we did this the first time, we said we didn't really want to solve this system of two equations in two unknowns, but we could have. So we can also model this using n of t equals a e to the power kt. So to solve this system of equations, the thing we might notice is that both equations have this factor of a on the left-hand side. And so if we divide the two equations, we'll be able to eliminate that common factor of a. So let's divide the larger by the smaller to avoid fractions. Well, actually, we can't avoid fractions once we move beyond a certain level of mathematics. And the common factor of a drops out, and we can use our rules of exponents to simplify the left-hand side. And we'll hit both sides with a log and solve, which gives us our value of k, and you'll notice the value of k is exactly the same no matter how we solved it. Now, since we know that n of 25 is equal to 8, we can substitute these in to find our value for a. And this time our value of a is very different, and that's because that we've incorporated this translation into that factor. And again, we can find out what happens after t equals 60, which is the same as we got before. And if we want to find when we have 100,000 people infected, we'll solve the equation n of t equal to 100, and get me get the same answer.