Up to this point, you’ve learned how to model linear and quadratic equations. The next relationship you’ll learn to model are exponential decay and growth, and they are mathematically represented in the form y = ab^x, where the variable x occurs as an exponent.

Exponential equations are used to model phenomena that increase or decrease slowly at first and then much more quickly. Take, for example, the growth of bacteria on a petri dish. Bacteria grow through a process known as binary fission, where the cytoplasm of a single parent cell splits into two progeny cells known as daughter cells (2 cells). In the next generation, the daughter cells also split via binary fission (4 cells), and the process continues until all available resources are consumed after x generations later. A table showing generations versus cell numbers are provided to your left. Therefore, exponential models are used to model things such as population, radioactive decay, or compound interest earned on a bank account.

In this chapter, you will focus mainly on situations where the output either grows or decays exponentially and we’ll eventually learn how to solve them using logarithms!

As mentioned already, an exponential function is one in which the independent variable (commonly the variable x) appears in the exponent position. The quantity that is raised to the power is called the base or growth factor.

Examples of exponential functions

When the exponent is positive (equations 1, 2, 3, and 4), the graph of the function increases exponentially, and when the it’s negative (equations 5 and 6), the graph decreases exponentially. Graphs of exponent equations provided below for clarity.

One of our goals this chapter is to model continuous growth or decay problems like the one discussed at the beginning about bacteria using exponential equations. In fact, most naturally occurring phenomena grow continuously. For example, bacteria will continue to grow over a 24 hours period, producing new bacteria which will also grow. The bacteria do not wait until the end of the 24 hours, and then all reproduce at once. To model such an equation, each equation will require a special irrational number, e, defined long ago by a Mathematician named Euler (pronounced oil’er) as the equations “growth factor”.

The best way to understand how this number is defined is to recall the basic idea behind compound interest. Compound interest is when a certain amount of money at the beginning and a percentage of your investment is added onto the initial amount after a defined period of time. The same percentage is added in every subsequent period, but the final amount increases faster each time because the interest earned is added onto the new initial amount. This can mathematically be modelled using the formula:

Where:

• a = initial investment ($)
• n = percentage rate offered (written as a decimal)
• m = number of periods (in 1 year)
• t = number of years

If you set a = $1, n = 100% → 1, and t = 1 gets you:

And if you were to set m to a very large number, say a million, you’d get an output close to ≈2.71828… The larger the m value, the longer the never ending decimal becomes. This suggests that as m grows infinity larger, or as the number of compounding periods increase continuously to infinity, the output y approaches the number 2.71828. Like π, ≈2.71828 is an irrational number; this means it cannot be represented as a fraction because the numbers after the decimal does not end. And like π, it gets its own special symbol e, which commemorates the name of the mathematician Euler.

The importance of this number comes whenever you’re looking to compute continuous growth or decay. For example, if you’re told that the growth of a population is continuous, the growth factor, b, is replaced with eⁿ, where ‘n’ represents the fixed rate per unit time and ‘t’ represents the elapsed time.

As mentioned before, when the exponent is positive, exponential growth occurs, and the opposite is true for when it’s negative.

Example of Continuous Exponential Growth

Certain bacteria, given favorable growth conditions, grow continuously at a rate of 4.6% a day. Find the bacterial population after thirty-six hours, if the initial population was 250 bacteria.

Solution:

We start with the formula:

$y=a{b}^x \leftrightarrow \boxed{y=a{b}^{nt}}$

Since you’re told that it’s continuous growth, change the base to e and make sure nt is positive:

$y=a{e}^{nt}$

The letter a represents the initial population, y represents the final population, n represents the rate, and t presents the time. Since the rate is given as a percentage per day, you need to write it as a decimal and convert days to hours so that when you multiply rate × time, the units cancel out:

$$\begin{gathered} \frac{4.6\%}{day}÷100\%\Rightarrow \frac{0.046}{\bcancel{day}}\times \frac{1 \bcancel{day}}{24 hours}\Rightarrow \boxed{\frac{0.046}{24 hour}} \\ x=\frac{0.046}{24 \bcancel{hour}}\times 36 \bcancel{hours}=0.069 \\ y=250e^{0.069} \\ y\cong 268 \end{gathered}$$

Therefore, after 36 hours of continuous growth, the final population (y) is 268 bacteria.

A second example of continuous exponential growth can be found below, but what’s interesting is that the word “continuous” isn’t written. The reason you can assume that it is continuous is because a fixed rate, given as a percentage per unit time, is given suggesting an instantaneous rate of change.

Non-continuous Exponent Equations

It’s important to note that not all growth or decay situations exhibit continuous growth. If the question does not mention “continuous” and a fixed percent isn’t provided, then the base needs to be replaced with the growth factor mentioned in the question instead. Watch the links below to see how to solve questions where it’s NOT a continuous model:

• Solve an exponential growth problem involving bacterial growth – Question 1
  • In case you’re curious as to why a different formula is used in this video, it’s actually similar to the one discussed above, but since a percentage rate isn’t given, writing it like that makes it easier to input the provided information. However, if you insist on using y = ab^nt (as before), you would have to set n as 1/8 days in decimal form or 12.5% / day in percentage form.
• Solve an exponential growth problem involving bacterial growth – Question 2