Question 13 (5 points)

Find a logarithmic function to model the data.

[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 60 \\
\hline
2 & 54 \\
\hline
3 & 51 \\
\hline
4 & 50 \\
\hline
5 & 46 \\
\hline
6 & 45 \\
\hline
7 & 44 \\
\hline
\end{array}
\][/tex]



Answer :

To model the data using a logarithmic function, we can use the form:

[tex]\[ y = a \ln(x) + b \][/tex]

where \( \ln(x) \) represents the natural logarithm of \( x \), and \( a \) and \( b \) are constants that we need to determine.

Given the data points:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 60 \\ \hline 2 & 54 \\ \hline 3 & 51 \\ \hline 4 & 50 \\ \hline 5 & 46 \\ \hline 6 & 45 \\ \hline 7 & 44 \\ \hline \end{array} \][/tex]

we need to find the best values for constants \( a \) and \( b \) that fit this logarithmic model to the given data points. After performing the necessary calculations and fitting the logarithmic model to the data, we determine the values of \( a \) and \( b \).

The fitted parameters are:

[tex]\[ a = -8.245225947626354 \][/tex]
[tex]\[ b = 60.04169738027974 \][/tex]

Therefore, the logarithmic function that models the provided data is:

[tex]\[ y = -8.245225947626354 \ln(x) + 60.04169738027974 \][/tex]

This function captures the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] based on the given data points.