Decay rate is a measure of how quickly a quantity, such as the amount of a radioactive substance or the amplitude of a vibrating system, decreases or diminishes over time. It is a fundamental concept in the study of exponential and logarithmic equations, which describe processes that exhibit exponential growth or decay.
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The decay rate is typically represented by the Greek letter $\lambda$ (lambda) and is the inverse of the half-life of a quantity.
Exponential decay follows the equation $y = y_0 e^{-\lambda t}$, where $y_0$ is the initial value, $t$ is time, and $e$ is the base of the natural logarithm.
The half-life of a quantity is related to the decay rate by the equation $t_{1/2} = \frac{\ln 2}{\lambda}$, where $t_{1/2}$ is the half-life.
Logarithmic equations can be used to solve for the decay rate or half-life of a quantity given other variables, such as the initial value and the value at a specific time.
Decay rate is an important concept in fields such as radioactive dating, population dynamics, and signal processing, where exponential and logarithmic relationships are observed.
Review Questions
Explain how the decay rate is related to the half-life of a quantity.
The decay rate, represented by the Greek letter $\lambda$, is inversely related to the half-life of a quantity. The half-life is the time it takes for a quantity to decrease to half of its initial value. The relationship between the decay rate and half-life is given by the equation $t_{1/2} = \frac{\ln 2}{\lambda}$, where $t_{1/2}$ is the half-life and $\lambda$ is the decay rate. This means that a higher decay rate corresponds to a shorter half-life, and vice versa.
Describe how the decay rate is used in the equation for exponential decay.
The decay rate, $\lambda$, is a key parameter in the equation for exponential decay, which is given by $y = y_0 e^{-\lambda t}$. In this equation, $y_0$ is the initial value of the quantity, $t$ is the time, and $e$ is the base of the natural logarithm. The decay rate, $\lambda$, determines the rate at which the quantity decreases over time. A higher decay rate results in a faster decrease in the quantity, while a lower decay rate leads to a slower decrease.
Explain how logarithmic equations can be used to solve for the decay rate or half-life of a quantity.
Logarithmic equations can be used to solve for the decay rate or half-life of a quantity that exhibits exponential decay. For example, if the initial value, $y_0$, and the value at a specific time, $y$, are known, the decay rate can be calculated using the equation $\lambda = -\frac{\ln(y/y_0)}{t}$. Similarly, if the decay rate is known, the half-life can be calculated using the equation $t_{1/2} = \frac{\ln 2}{\lambda}$. These logarithmic relationships allow for the determination of key parameters related to the decay process.
Related terms
Half-Life: The time it takes for a quantity to decrease to half of its initial value, which is inversely related to the decay rate.
A process where a quantity decreases by a constant proportion over equal intervals of time, resulting in an exponential relationship between the quantity and time.
Equations that describe relationships between quantities that exhibit exponential growth or decay, and can be used to solve for unknown variables related to the decay rate.