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 are tasked with finding a logarithmic function to model the data. Let's denote this function as [tex]\( f(x) \)[/tex].
Logarithmic functions typically take the form:
[tex]\[
f(x) = a - b \ln(x)
\][/tex]
We need to determine the coefficients [tex]\( a \)[/tex] and [tex]\( b \)[/tex] that best fit the given data. The analysis has provided us with:
[tex]\[
a \approx 60.04 \\
b \approx 8.25
\][/tex]
Therefore, the logarithmic function that models the given data is:
[tex]\[
f(x) = 60.04 - 8.25 \ln(x)
\][/tex]
Upon inspection of the choices provided:
- [tex]\( f(x) = 60.73(0.95)^x \)[/tex] is an exponential decay function and does not fit our form.
- [tex]\( f(x) = 0.93(60.73)^x \)[/tex] is also an exponential growth function and does not fit our form.
- [tex]\( f(x) = 8.25 - 60.04 \ln(x) \)[/tex] is not consistent with our derived coefficients, as the coefficients are swapped and hence also incorrect.
- [tex]\( f(x) = 60.04 - 8.25 \ln(x) \)[/tex] exactly matches our derived logarithmic function.
Given this, the correct logarithmic function modeling the given data is:
[tex]\[
f(x) = 60.04 - 8.25 \ln(x)
\][/tex]
