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Chapter 5: Problem 19

Determine whether an exponential, power, or logarithmic model (or none or several of these) is appropriate for the data by determining which (if any) of the following sets of points are approximately linear: $$\\{(x, \ln y)\\}, \quad\\{(\ln x, \ln y)\\}, \quad\\{(\ln x, y)\\}$$ where the given data set consists of the points \(\\{(x, y)\\}\) $$\begin{array}{|l|c|c|c|c|c|c|} \hline x & 5 & 10 & 15 & 20 & 25 & 30 \\ \hline y & 17 & 27 & 35 & 40 & 43 & 48 \\ \hline \end{array}$$

Short Answer

Expert verified
Answer: The logarithmic model is the most appropriate for modeling the given data set because, among the three transformed sets, it appears to have the most linear pattern.

Step by step solution

01

Transform the given data set using the three models

For each data point \((x, y)\), transform it into the three models: 1. \((x, \ln y)\) for the exponential model 2. \((\ln x, \ln y)\) for the power model 3. \((\ln x, y)\) for the logarithmic model
02

Analyze the transformed sets for linearity

Check if any of the transformed sets are approximately linear. If any set is linear, then the corresponding model is appropriate for the data set.
03

Determine the appropriate model for the data set

If a linear relationship is found in any of the transformed sets, conclude which model (exponential, power, or logarithmic) is appropriate for modeling the given data set. Step 1 results: Given the data set, we will transform it into the three models: 1. Exponential model: \(\\{(x, \ln y)\\} = \\{(5, \ln 17), (10, \ln 27), (15, \ln 35), (20, \ln 40), (25, \ln 43), (30, \ln 48)\\}\) 2. Power model: \(\\{(\ln x, \ln y)\\} = \\{(\ln 5, \ln 17), (\ln 10, \ln 27), (\ln 15, \ln 35), (\ln 20, \ln 40), (\ln 25, \ln 43), (\ln 30, \ln 48)\\}\) 3. Logarithmic model: \(\\{(\ln x, y)\\} = \\{(\ln 5, 17), (\ln 10, 27), (\ln 15, 35), (\ln 20, 40), (\ln 25, 43), (\ln 30, 48)\\}\) Step 2 analysis: Visually inspecting the three transformed sets, none of them appear to be perfectly linear. However, the set \(\\{(\ln x, y)\\}\) seems to resemble a more linear pattern than the other two sets. Step 3 conclusion: Considering the step 2 analysis, none of the sets appears to be perfectly linear. However, since the logarithmic model seems to be the most linear among the three, it could be considered the most appropriate model for the given data set.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Models
Exponential models are powerful tools for describing growth or decay processes. In mathematical terms, an exponential model is expressed as \( y = a \times e^{bx} \), where \( a \) and \( b \) are constants, and \( e \) is the base of the natural logarithm.

These models are suitable for data that shows hockey-stick-like growth, meaning they grow slowly at first and then rapidly, or they can describe decay processes like radioactive decay.
  • To determine if a dataset fits an exponential model, we translate the points to the form \((x, \ln y)\).
  • If the transformed data creates a linear pattern, then the exponential model is appropriate. This means that the variable's rate of increase is proportional to its current value.


In our exercise, the transformation to \((x, \ln y)\) does not show a strong linear pattern. Therefore, an exponential model might not be the best fit for this data.
Power Models
Power models are another type of mathematical modeling, used when the relationship between variables is best described by an equation of the form \( y = a \times x^b \).

This model is particularly useful in scenarios where quantities are related by powers, such as physical laws like gravity or electrical circuits. It is suitable when data shows a consistent proportional relationship on a log-log scale.
  • To check for a power model fit, transform the data into \((\ln x, \ln y)\).
  • If the transformed data appears linear, a power model is likely suitable for the data.


In the case of our exercise, transforming the data with \((\ln x, \ln y)\) did not result in a clear linear pattern. Hence, the power model doesn't suit the dataset provided.
Logarithmic Models
Logarithmic models are useful when data changes quickly at first and then levels off. These can be described with a logarithmic equation of the form \( y = a + b \ln x \), where \( a \) and \( b \) are constants.

Logarithmic models find applications in various sciences and financial fields, where initial rapid changes taper off. To analyze whether a logarithmic model fits a dataset, we transform the data into \((\ln x, y)\) format.
  • If this transformation results in a linearly arranged data set, it indicates that a logarithmic model is a suitable approach.


In our exercise scenario, the transformed set \((\ln x, y)\) was observed to be closer to a linear arrangement compared to other models. Thus, a logarithmic model would be the most appropriate choice to describe the relationship in the given dataset.

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Most popular questions from this chapter

Assume that you watched 1000 hours of television this year, and will watch 750 hours next year, and will continue to watch \(75 \%\) as much every year thereafter. (a) In what year will you be down to ten hours per year? (b) In what year would you be down to one hour per year?

This exercise provides a justification for the claim that the function \(M(x)=c(.5)^{x / h}\) gives the mass after \(x\) years of a radioactive element with half-life \(h\) years. Suppose we have \(c\) grams of an element that has a half-life of 50 years. Then after 50 years, we would have \(c\left(\frac{1}{2}\right)\) grams. After another 50 years, we would have half of that, namely, \(c\left(\frac{1}{2}\right)\left(\frac{1}{2}\right)=c\left(\frac{1}{2}\right)^{2}\) (a) How much remains after a third 50-year period? After a fourth 50 -year period? (b) How much remains after \(t 50\) -year periods? (c) If \(x\) is the number of years, then \(x / 50\) is the number of 50-year periods. By replacing the number of periods \(t\) in part (b) by \(x / 50,\) you obtain the amount remaining after \(x\) years. This gives the function \(M(x)\) when \(h=50\) The same argument works in the general case (just replace 50 by \(h\) ). Find \(M(x)\).

(a) Graph \(f(x)=x^{5}\) and explain why this function has an inverse function. (b) Show algebraically that the inverse function is \(g(x)=x^{1 / 5}\) (c) Does \(f(x)=x^{6}\) have an inverse function? Why or why not?

Approximating Logarithmic Functions by Polynomials. For each positive integer \(n,\) let \(f_{n}\) be the polynomial function whose rule is $$ f_{n}(x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\frac{x^{5}}{5}-\cdots \pm \frac{x^{n}}{n} $$ where the sign of the last term is \(+\) if \(n\) is odd and \(-\) if \(n\) is even. In the viewing window with \(-1 \leq x \leq 1\) and \(-4 \leq y \leq 1,\) graph \(g(x)=\ln (1+x)\) and \(f_{4}(x)\) on the same screen. For what values of \(x\) does \(f_{4}\) appear to be a good approximation of \(g ?\)

The population of St. Petersburg, Florida (in thousands) can be approximated by the function $$ g(x)=-127.9+81.91 \ln x \quad(x \geq 70) $$ where \(x=70\) corresponds to 1970 (a) Estimate the population in 1995 and 2003 . (b) If this model remains accurate, when will the population be \(260,000 ?\)
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