For the following exercises, use this scenario: The...

For the following exercises, use this scenario: The population $P$ of an endangered species habitat for wolves is modeled by the function $P(x)=\frac{558}{1+54.8 e^{-0.462 x}},$ where $x$ is given in years.
How many years will it take before there are 100 wolves in the habitat?

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Key Concepts

Logistic Growth Model

A logistic growth model is used to describe populations that grow rapidly initially but then slow down as they approach a maximum limit or carrying capacity. This type of model is characterized by an S-shaped curve and incorporates factors that limit growth as the population becomes large.

Exponential Functions

Exponential functions represent processes where a quantity increases or decreases at a rate proportional to its current value. They are fundamental in modeling the rapid change observed at the start of logistic growth before other moderating factors take effect.

Natural Logarithms

Natural logarithms are the inverse functions of exponential functions with base e, and they are instrumental in solving equations where the unknown is in the exponent. They allow for the linearization of exponential equations, making it possible to isolate the variable of interest.

Solving Exponential Equations

Solving exponential equations involves manipulating equations to isolate the exponent, often by taking logarithms on both sides. This process is essential in determining the time required for a given population level or in finding other growth-related parameters in models.

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