To find the logarithmic regression model of the form [tex]\( y = a + b \ln(x) \)[/tex] for the given data set, we can follow these steps:
1. Define the Data:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
9 & 5 \\
12 & 17 \\
22 & 45 \\
40 & 60 \\
\hline
\end{array}
\][/tex]
2. Apply Logarithmic Transformation:
Define [tex]\( u = \ln(x) \)[/tex] where [tex]\( \ln \)[/tex] denotes the natural logarithm. Compute [tex]\( u \)[/tex] for each value of [tex]\( x \)[/tex].
[tex]\[
\begin{array}{|c|c|c|}
\hline
x & \ln(x) & y \\
\hline
9 & \ln(9) & 5 \\
12 & \ln(12) & 17 \\
22 & \ln(22) & 45 \\
40 & \ln(40) & 60 \\
\hline
\end{array}
\][/tex]
Which gives us approximately:
[tex]\[
\begin{array}{|c|c|c|}
\hline
x & \ln(x) & y \\
\hline
9 & 2.1972 & 5 \\
12 & 2.4849 & 17 \\
22 & 3.0910 & 45 \\
40 & 3.6889 & 60 \\
\hline
\end{array}
\][/tex]
3. Perform Linear Regression on the Transformed Data:
We now perform a linear regression on the data points [tex]\((\ln(x), y)\)[/tex]:
[tex]\[
\begin{array}{|c|c|}
\hline
\ln(x) & y \\
\hline
2.1972 & 5 \\
2.4849 & 17 \\
3.0910 & 45 \\
3.6889 & 60 \\
\hline
\end{array}
\][/tex]
We fit a line [tex]\(y = a + b \cdot \ln(x)\)[/tex] to this transformed data.
4. Find the Coefficients [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
Using the least squares method (or another suitable method for linear regression), we solve for the coefficients [tex]\(a\)[/tex] and [tex]\(b\)[/tex] that best fit the transformed data.
From the calculations and fitting, we obtain the coefficients:
[tex]\[
a = -76.20384525969956
\][/tex]
[tex]\[
b = 37.67347576983646
\][/tex]
5. Construct the Model:
Using the coefficients found, the logarithmic regression model is:
[tex]\[
y = -76.20384525969956 + 37.67347576983646 \ln(x)
\][/tex]
So, the regression equation modeling the height of the stalk of corn as a function of the days is:
[tex]\[
y = -76.20384525969956 + 37.67347576983646 \ln(x)
\][/tex]
Thus, we have:
[tex]\[
\begin{array}{l}
a = -76.20384525969956 \\
b = 37.67347576983646
\end{array}
\][/tex]
