Lesson 7 Logs Go Viral Practice Understanding

As we learned in the previous lesson, Compounding the Problem, when money is compounded continuously, it turns out that the base of the exponential growth function is . It is true that basically all situations that grow or decay continuously can be modeled with a base exponential function. The basic formulas are:

Continuous growth:

Continuous decay:

In both cases, is the amount of “stuff” at time . is the amount of “stuff” to start with, or at , and is the rate of growth or decay. So, if we’re talking about money, and are dollar amounts. If we’re talking about a population, and are the number of people in the population. If we’re talking about radioactive decay, then and are the mass of the radioactive substance. In other words, is what we started with and is what we have after it grows or decays.

Since there are many things that grow or decay continuously in nature, is called the natural exponential function and the base logarithm is called the natural logarithm. Think about the growth of bacteria. One cell splits into two cells. The two cells begin to split, and the resulting cells split and so on. In a group of many cells, there would be a cell splitting nearly continuously, thus it is a base exponential function. So, to use this base exponential and its inverse, imagine the following scenarios:

You are an epidemiologist, a person who studies the outbreak and spread of diseases. Part of your job is to help avoid a pandemic—a worldwide outbreak of a disease. You know some of the most difficult diseases to deal with are viruses because they don’t respond to many of the medicines we have available and because viruses are able to mutate and change quickly, making it more difficult to contain them. You have been studying a new virus that causes people to break out in spots. Suddenly, a colleague rushes into your office to inform you there is a confirmed outbreak of the virus in Europe. The growth of the virus through a population is continuous (until it is somehow contained) at a rate of per day. The current outbreak has confirmed victims.

Now you have received a report of a mysterious illness that seems to turn the infected humans into mindless zombies has broken out in a major American city. Since the hungry zombies prey upon innocent people, the outbreak grows continuously at a rate of per day. The outbreak begins with .

Now we’re going to get a little more far-fetched in the scenario. Let’s say that zombies produce radioactive goo that decays continuously with a half-life of . (That’s one more danger of having zombies around.) The half-life tells us that after , only half of the amount of goo we started with is remaining.