Law of Exponential Change

Definition

The Law of Exponential Change states that if a quantity $y$ changes at a rate proportional to its current size, it follows the model: $y = y_0 e^{kt}$ where $y_0$ is the initial amount and $k$ is the rate constant.

• How to read: “The quantity y equals y zero times e to the power k t.”
• Meaning: Growth or decay proportional to current size produces exponential curves—$k>0$ grows, $k<0$ decays.

Why It Matters

Growth is not always linear. The law of exponential change dictates the “explosive” nature of everything from compound interest to biological growth and technology, warning us that “doubling” can quickly overwhelm any system.

Core Concepts

• Rate Proportionality: Defined by the differential equation $\frac{dy}{dt} = ky$.

  • How to read: “The derivative of y with respect to t equals k times y.”
  • Meaning / when to use: The rate of change is a fixed fraction of the current amount—the defining ODE whose solution is $y = y_0 e^{kt}$.”

• Growth vs. Decay: If $k > 0$, the quantity increases (growth); if $k < 0$, it decreases (decay).

• Constant Ratio: Over any fixed time interval, the quantity changes by the same percentage or ratio, regardless of the starting amount.

Connected Concepts

• Exponential Equations
• Logarithmic Equations
• Exponential Growth Model
• Exponential Decay Model
• Law of Sines