Exponential Decay

Definition

Exponential decay is a mathematical model that describes the gradual decrease of a quantity over time. It is characterized by an initial value that diminishes at a constant proportional rate, resulting in an exponential decline.

5 Must Know Facts For Your Next Test

1. Exponential decay is often used to model the behavior of radioactive materials, the concentration of certain drugs in the body, and the depreciation of assets over time.
2. The rate of exponential decay is determined by the decay constant, which represents the fraction of the remaining quantity that is lost in each time interval.
3. Exponential decay functions have the form $f(t) = a \cdot e^{-bt}$, where $a$ is the initial value, $b$ is the decay constant, and $e$ is the base of the natural logarithm.
4. Logarithmic functions are used to represent the growth or decay rate of an exponential process, as they are the inverse of exponential functions.
5. Geometric sequences exhibit exponential growth or decay, where each term is a constant multiple of the previous term.

Review Questions

• Explain how exponential decay is represented in the context of evaluating and graphing exponential functions (10.2).
  • In the context of evaluating and graphing exponential functions (10.2), exponential decay is represented by a function of the form $f(t) = a \cdot e^{-bt}$, where $a$ is the initial value, $b$ is the decay constant, and $e$ is the base of the natural logarithm. The graph of an exponential decay function starts at the initial value $a$ and decreases asymptotically towards the x-axis as time $t$ increases. The rate of decay is determined by the value of the decay constant $b$, with larger values of $b$ indicating a faster rate of decay.
• Describe how exponential decay is related to the evaluation and graphing of logarithmic functions (10.3).
  • Logarithmic functions are the inverse of exponential functions, and they can be used to represent the growth or decay rate of an exponential process. In the context of evaluating and graphing logarithmic functions (10.3), exponential decay is related to the logarithmic function in the sense that the logarithm of an exponentially decaying quantity will result in a linear function. This relationship allows for the analysis of exponential decay processes using logarithmic functions, which can provide insights into the rate of decay and the underlying behavior of the system.
• Analyze how exponential decay is applied in the context of solving exponential and logarithmic equations (10.5).
  • When solving exponential and logarithmic equations (10.5), exponential decay is a key concept that is often utilized. Exponential decay equations, such as $f(t) = a \cdot e^{-bt}$, can be transformed into logarithmic equations by applying the properties of logarithms. This allows for the determination of unknown parameters, such as the initial value $a$ or the decay constant $b$, based on the given information. Furthermore, the understanding of exponential decay is essential in solving real-world problems involving the behavior of systems that exhibit exponential decay, such as radioactive decay, population dynamics, or the depreciation of assets.

Related terms

Exponential Function: A function that grows or decays exponentially, where the rate of change is proportional to the current value.

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

Geometric Sequence: A sequence where each term is a constant multiple of the previous term, exhibiting exponential growth or decay.