Exponential Decay

Definition

Exponential decay is a mathematical function that describes a quantity decreasing at a rate proportional to its current value. It is a fundamental concept in various fields, including physics, chemistry, biology, and finance, and is closely related to the properties of exponential functions and logarithmic functions.

5 Must Know Facts For Your Next Test

1. Exponential decay is characterized by a constant rate of change, where the rate of decrease is proportional to the current value of the quantity.
2. The general form of an exponential decay function is $f(t) = a ext{ } ullet ext{ } e^{-kt}$, where $a$ is the initial value, $k$ is the decay rate, and $t$ is the independent variable (usually time).
3. The half-life of an exponentially decaying quantity is the time it takes for the quantity to decrease to half of its initial value, and is given by the formula $t_{1/2} = ext{ln}(2) / k$.
4. Exponential decay is often used to model the behavior of radioactive decay, population growth and decline, and the depreciation of assets.
5. Logarithmic functions are used to analyze and interpret exponential decay processes, as they can linearize the exponential relationship and provide a way to determine the decay rate.

Review Questions

• Explain how the concept of exponential decay is related to the properties of exponential functions.
  • Exponential decay is a specific type of exponential function, where the function decreases over time at a constant rate. The general form of an exponential decay function is $f(t) = a ext{ } ullet ext{ } e^{-kt}$, where $a$ is the initial value, $k$ is the decay rate, and $t$ is the independent variable (usually time). The negative exponent in the function indicates that the value is decreasing over time, rather than increasing as in a typical exponential growth function.
• Describe how the concept of half-life is used to analyze exponential decay processes.
  • The half-life of an exponentially decaying quantity is the time it takes for the quantity to decrease to half of its initial value. This is a key characteristic of exponential decay, and the formula for half-life is $t_{1/2} = ext{ln}(2) / k$, where $k$ is the decay rate. By knowing the half-life of a decaying quantity, you can determine the rate at which it is decreasing and make predictions about its future behavior. Half-life is commonly used in fields like radioactive decay, population dynamics, and the depreciation of assets.
• Explain how logarithmic functions are used to interpret and analyze exponential decay processes.
  • Logarithmic functions are the inverse of exponential functions, and they can be used to linearize exponential decay relationships. By taking the natural logarithm of an exponential decay function, you can transform the equation into a linear form, $ ext{ln}(f(t)) = ext{ln}(a) - kt$, where the slope of the line is equal to the decay rate $k$. This allows for easier interpretation and analysis of exponential decay processes, as the logarithmic transformation provides a way to determine the decay rate and make predictions about the future behavior of the decaying quantity.

Related terms

Exponential Function: A function of the form $f(x) = a ext{ } ullet ext{ } b^x$, where $a$ is the initial value and $b$ is the base of the exponential function.

Half-life: The time it takes for a quantity to decrease to half of its initial value, which is a key characteristic of exponential decay.

Logarithmic Function: The inverse function of an exponential function, used to represent the rate of change in an exponential decay process.