In this explainer, we will learn how to solve real-world problems involving exponential functions.
Recall that the basic exponential function is given by where the base is a positive number other than 1. The general form is and, in “modeling” real phenomena, we have
- Independent variable is usually time.
- Since , the quantity is the initial value of what the function is measuring. The value at time .
- The base tells us something about the “rate” at which our quantity is changing with time. Growth will correspond to a base , while decay will be when .
Assuming a positive value for , the graphs all look like this:
Note that is the amount after one unit of time. Notice also how the quantity changes from to , depending on how compares to 1. Since , we see that the values at times are in the geometric sequence with common ratio . Thus, “doubling” every unit of time while “halving” .
Here is an example.
Example 1: Forming an Expression to Model Real-World Exponential Growth
The number of bacteria in a laboratory quadruples every hour. There were initially 200 bacteria. Write an expression for , the number of bacteria hours after the initial measurement.
This is an example of exponential growth, so for some constants where independent variable time is measured in hours. The quadrupling means , and being the initial amount gives , so
In many cases, the rate is given as a “percentage rate.” Recall what this means. Suppose that the value of a car depreciates by each year. This means that if the value was at the beginning of the year, then, at the end, the value is less by of this amount and is
If expresses this depreciation, then
Note that this rate is less than 1, as expected of a decaying process. Using percentage rates is talking about rates relative to 1, so a growth rate of means a rate of .
Here is another example.
Example 2: Identifying the Initial Value and the Rate of Increase of an Exponential Growth Function
The value, dollars, of a property years from now can be modeled by the function
- What is the property’s value now?
- What is the rate of increase in the property’s value?
- The property’s value now is
- In the form , we see that and . Writing and solving for gives a percentage rate of increase in value of So, the property’s value is increasing at a rate of every year.
Banks often accrue interest on a periodic basis. For example if a “nominal” annual rate of is collected every three months. This means that, in a single year, it is gathered at the rate of four times in that year. So, starting with a loan of dollars, the total debt at year end is
Since , this represents a debt of which is what would be levied if the annual rate was actually , levied just the one time. More generally, a (nominal) rate of percent over periods per year starting with an amount will grow, after years, to .
Example 3: Making Calculation with Compound Interest
If $800 is earning interest semiannually at per annum, what is the amount after years?
The earnings are over periods at a nominal rate of with dollars. The formula for the amount after a single year is after years
The effective (or annual) rate is .
Since exponential growth/decay is modeled with exponential functions, we can solve for the independent variable (say time) using logarithms.
Example 4: Solving Real-World Problems Involving Exponential Growth
A microorganism reproduces by binary fission, where every hour each cell divides into two cells. Given that there are 24,431 cells to begin with, determine how long it will take for there to be 97,724 cells.
The number of cells , after hours, is growing by where and the rate because of the “doubling.” We must solve for : which gives
So, it will take 2 hours for the cell population to reach 97,724.
Example 5: Interpreting Parameters in Exponential Functions in a Real-World Context
The number of Ebola infections in West Africa at the start of an epidemic followed an exponential growth. It is given by , with the number of days after the first infection.
- What does the coefficient 0.075 represent?
- It is the number of new infections per day.
- It is the time it takes for the number of infections to be multiplied by .
- is the time it takes for the number of infections to be multiplied by .
- It is the percentage of the daily growth in the number of infections .
- It is the number of days after the first infection.
- By rewriting the formula in the form , find the percentage of the daily growth in the number of infections. Give your answer to one decimal place.
- We will discuss each of these options in turn.
Option (A) does not seem reasonable: an exponential growth will not have a constant rise in the number of infections. This answer would fit a linear growth.
(B) sounds better, since exponential figures grow multiplicatively. But this is not right. In a “doubling” formula, we get the term which says that time is when the quantity is multiplied by 2. This would be true if the formula read instead.
(D) is saying that the daily rate of increase of the infected is . This would correspond to a formula involving a term like .
(E) is not very meaningful.
(C) is correct in the same way that (B) was incorrect: setting in the formula gives , and when , we get . Each interval of this amount of time multiplies the infections by .
The right answer is option (C).
- We rewrite as follows:
When the base is written as , the percentage rate of growth is exactly which is