Exponential Function and Decay

In mathematics, exponential decay describes the process of reducing an amount by a consistent percentage rate over a period of time. It can be expressed by the formula y=a(1-b)wherein y is the final amount, a is the original amount, b is the decay factor, and x is the amount of time that has passed.

The exponential decay formula is useful in a variety of real world applications, most notably for tracking inventory that's used regularly in the same quantity (like food for a school cafeteria) and it is especially useful in its ability to quickly assess the long-term cost of use of a product over time.

Exponential decay is different from linear decay in that the decay factor relies on a percentage of the original amount, which means the actual number the original amount might be reduced by will change over time whereas a linear function decreases the original number by the same amount every time.

It is also the opposite of exponential growth, which typically occurs in the stock markets wherein a company's worth will grow exponentially over time before reaching a plateau. You can compare and contrast the differences between exponential growth and decay, but it's pretty straightforward: one increases the original amount and the other decreases it.

Elements of an Exponential Decay Formula

To start, it's important to recognize the exponential decay formula and be able to identify each of its elements:

y = a (1-b)x

In order to properly understand the utility of the decay formula, it is important to understand how each of the factors is defined, beginning with the phrase "decay factor"—represented by the letter b in the exponential decay formula—which is a percentage by which the original amount will decline each time.

The original amount here—represented by the letter in the formula—is the amount before the decay occurs, so if you're thinking about this in a practical sense, the original amount would be the amount of apples a bakery buys and the exponential factor would be the percentage of apples used each hour to make pies.

The exponent, which in the case of exponential decay is always time and expressed by the letter x, represents how often the decay occurs and is usually expressed in seconds, minutes, hours, days, or years.

An Example of Exponential Decay

Use the following example to help understand the concept of exponential decay in a real-world scenario:

On Monday, Ledwith’s Cafeteria serves 5,000 customers, but on Tuesday morning, the local news reports that the restaurant fails health inspection and has—yikes!—violations related to pest control. Tuesday, the cafeteria serves 2,500 customers. Wednesday, the cafeteria serves only 1,250 customers. Thursday, the cafeteria serves a measly 625 customers.

As you can see, the number of customers declined by 50 percent every day. This type of decline differs from a linear function. In a linear function, the number of customers would decline by the same amount every day. The original amount (a) would be 5,000, the decay factor (b ) would, therefore, be .5 (50 percent written as a decimal), and the value of time (x) would be determined by how many days ​Ledwith wants to predict the results for.

If Ledwith were to ask about how many customers he would lose in five days if the trend continued, his accountant could find the solution by plugging all of the above numbers into the exponential decay formula to get the following: ​

y = 5000(1-.5)5

The solution comes out to 312 and a half, but since you can't have a half customer, the accountant would round the number up to 313 and be able to say that in five days, Ledwith could expect to lose another 313 customers!

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Ledwith, Jennifer. "Exponential Function and Decay." ThoughtCo, Jan. 29, 2020, thoughtco.com/exponential-decay-definition-2312215. Ledwith, Jennifer. (2020, January 29). Exponential Function and Decay. Retrieved from https://www.thoughtco.com/exponential-decay-definition-2312215 Ledwith, Jennifer. "Exponential Function and Decay." ThoughtCo. https://www.thoughtco.com/exponential-decay-definition-2312215 (accessed March 21, 2023).