Modeling Real-World Exponentials

Exponential growth occurs when a quantity increases by a fixed percentage over equal time intervals. In finance, this is called Compound Interest. Unlike simple interest, where you only earn on your initial deposit, compound interest allows you to earn interest on your interest. The standard formula is $A = P(1 + r)^t$, where $A$ is the final amount, $P$ is the principal (starting amount), $r$ is the interest rate (expressed as a decimal), and $t$ is time. The term $(1 + r)$ is known as the growth factor. If the growth factor is greater than 1, the value will skyrocket over time as the base continues to multiply against itself.

Not everything grows. Depreciation describes how items like cars or smartphones lose value over time. Similarly, half-life describes how radioactive substances break down. These are examples of Exponential Decay. The formula is almost identical to growth: $y = a(1 - r)^t$. The only difference is the minus sign! Here, the decay factor $(1 - r)$ is always between 0 and 1. For example, if a car loses 15% of its value each year, it retains 85% of its value. Therefore, your decay factor is $0.85$. Even though the value drops quickly at first, it slows down as the total value gets smaller, never quite reaching zero.

Half-life is a specific type of exponential decay used in science. It is the time required for a quantity to reduce to exactly one-half of its initial value. Instead of a percentage rate $r$, we use a constant decay factor of $0.5$. The formula is often written as $A = P(0.5)^n$, where $n$ is the number of half-lives that have passed. To find $n$, you divide the total time by the length of one half-life: $n = \\frac{total\ time}{half-life\ period}$. This allows us to model everything from carbon dating ancient bones to how long medicine stays in your system.