The decay factor in exponential decay is a constant ratio that indicates how a quantity diminishes over equal time intervals.

More specifically, exponential decay refers to a process in which a quantity decreases at a rate that is proportional to its current value. This phenomenon is commonly observed in various contexts, such as radioactive decay, the cooling of objects, and the depreciation of assets. The decay factor plays a pivotal role in this process. It is a value between $0$ and $1$ that signifies the fraction of the quantity that remains after each time period. For instance, if a substance has a decay factor of $0.8$, it implies that $80\%$ of the substance remains after each time interval, indicating that $20\%$ has decayed.

Mathematically, if we denote the initial amount as $A_0$ and the decay factor as $b$, the amount $A$ remaining after $t$ time periods can be calculated using the formula:

In this equation, $b$ represents the decay factor, and $t$ denotes the number of time periods. This formula effectively illustrates how the quantity decreases exponentially over time.

Understanding the decay factor is essential for predicting the rate at which a quantity will diminish. For example, if you know the decay factor of a radioactive substance, you can estimate how much of it will remain after a specified number of years. This concept is invaluable not only in scientific contexts but also in finance, where it can be employed to model the depreciation of assets or the decline in value of investments over time.