Imagine a material has a half-life of 3 hours. This means that the amount of material left after 3 hours is half of what there was when it started. For example, if there are 200 milligrams of the material at noon, then at 3 o’clock there will be 100 milligrams left, and at 6 o’clock there will be 50 milligrams left.

If a scientist has 200 mg of the material, then the amount of material, in mg, can be modeled by the function  . In this model, represents a unit of time. Notice that the 200 represents the initial amount of material the scientist has. The number indicates that for every 1 unit of time, the amount of material is cut in half. Because the half-life is 3 hours, this means that must measure time in groups of 3 hours.

But what if we wanted to find the amount of material the scientist has each hour after taking it? We know there are 3 equal groups of 1 hour in a 3-hour period. We also know that because the material decays exponentially, it decays by the same factor in each of those intervals. In other words, if is the decay factor for each hour, then , or  . This means that over each hour, the medicine must decay by a factor of , which can also be written as . So if  is time in hours since the scientist had 200 mg of the material, we can express the amount of material in mg, , the scientist has as , or .