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Chapter 3: Problem 126

Logarithmic models are well suited to phenomena in which growth is initially rapid but then begins to level off. Describe something that is changing over time that can be modeled using a logarithmic function.

Short Answer

Expert verified
A possible real-world phenomenon that can be modeled with a logarithmic function is the spread of a disease, such as a virus, within a population. The corresponding model could be \(I(t) = P \cdot \log(t+1)\), where \(I(t)\) is the number of infected persons, \(P\) is the total population, and \(t\) is the time since the start of the outbreak.

Step by step solution

01

Understanding the behavior of logarithmic functions

Logarithmic functions, when graphed, show a growth that is initially rapid but begins to level off over time. This is because logarithms are the inverse of exponential functions. As an input gets larger, the rate of increase of the logarithm gets smaller.
02

Thinking of a relevant example

A good example of something that can be modeled with a logarithmic function is the way a certain disease, such as a virus, spreads in a population. In the early stages of an outbreak, the number of infected individuals can increase rapidly. However, as more and more people become infected, the rate of new infections starts to level off, since there are fewer and fewer individuals left to infect. This is true until measures are taken to control the spread or a substantial proportion of population develops immunity. Observing the graph in such scenarios, we can see that the growth can be represented by a logarithm.
03

Formulating the logarithmic model

Let's denote the total population as P, the number of infected persons as I, and the time as t. A possible model for the situation could be \(I(t) = P \cdot \log(t+1)\). This model indicates that the number of infected people I at any given time t is proportional to the logarithm of time (with a shift of 1 to avoid undefined values for \(t=0\)). The number of infected people is eventually limited by the total population P, which is why this factor is included in the model.

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