Suppose that a quantity y has an exponential growth model...

Suppose that a quantity $y$ has an exponential growth model
$y=y_{0} e^{k t}$ or an exponential decay model $y=y_{0} e^{-k t}$, and it is known that $y=y_{1}$ if $t=t_{1}$. In each case find a formula for $k$ in terms of $y_{0}, y_{1},$ and $t_{1},$ assuming that $t_{1} \neq 0$.

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Key Concepts

Exponential Growth and Decay Models

These models describe processes where quantities increase or decrease at rates proportional to their current values, typically expressed in the form y = y?e^(±kt). The positive exponent represents growth, while the negative exponent represents decay. Understanding these models is crucial for representing real-world phenomena such as population growth, radioactive decay, and interest calculations.

Solving Exponential Equations

To determine unknown constants in exponential equations, one often needs to isolate the exponential term and solve for the variable in the exponent. This involves substituting known values into the model and applying algebraic manipulation to express the equation in a form that allows for the extraction of the exponent, which in this context is the constant k.

Natural Logarithms

The natural logarithm is the inverse function of the exponential function with base e. It is an essential tool for solving equations where the variable of interest is in the exponent. By taking the natural logarithm of both sides of an equation, one can 'bring down' the exponent (using the property ln(e^x) = x), thereby converting the problem into a linear form suitable for further algebraic manipulation.

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