Using Logarithmic Regression
Examine the data below for a stalk of corn.
| Day, $x$ | 9 | 12 | 22 | 40 |
| :-------- | :- | :- | :- | :- |
| Height, $y$ (in) | 5 | 17 | 45 | 60 |
Use logarithmic regression to find an equation of the form [tex]$y=a+b \ln (x)$[/tex] to model the data.
[tex]a=[/tex]
[tex]b=[/tex]
Answer (2)
To model the height of a corn stalk over days using logarithmic regression, we determined that the values for the constants are approximately a = -76.20 and b = 37.67. Therefore, the equation that describes this relationship is y = -76.20 + 37.67 ln(x). This model allows predictions of corn stalk height based on the number of days since planting.
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Perform logarithmic regression on the given data points.
Calculate the values of a and b in the equation y = a + b ln ( x ) .
The calculated values are approximately a = − 76.20 and b = 37.67 .
The logarithmic equation that models the data is y = − 76.20 + 37.67 ln ( x ) . Therefore, the final answer is a = − 76.20 , b = 37.67 .
Explanation
Understanding the Problem We are given a set of data points ( x , y ) representing the height of a corn stalk on different days. Our goal is to find a logarithmic equation of the form y = a + b ln ( x ) that models this data. This means we need to determine the values of the constants a and b that best fit the given data.
Logarithmic Regression To find the values of a and b , we will perform a logarithmic regression on the given data. Logarithmic regression is a type of regression analysis used when the relationship between the independent variable x and the dependent variable y is logarithmic. In this case, we want to model the height y as a function of the natural logarithm of the day x .
Calculating a and b Using the data points (9, 5), (12, 17), (22, 45), and (40, 60), we can calculate the values of a and b using the formulas derived from the least squares method. After performing the calculations (which can be done using a calculator or statistical software), we find that:
a ≈ − 76.20 b ≈ 37.67
The Logarithmic Equation Therefore, the logarithmic equation that models the data is approximately:
y = − 76.20 + 37.67 ln ( x )
Final Answer The values of a and b are:
a = − 76.20 b = 37.67
Examples
Logarithmic regression is useful in various fields, such as environmental science, where it can model the relationship between pollution levels and distance from a source. In business, it can be used to model the relationship between advertising expenditure and sales. For example, if a company spends more on advertising, the sales might increase, but the rate of increase might slow down as the expenditure increases, which can be modeled using a logarithmic equation. This helps in making informed decisions about resource allocation and predicting future trends.
