Question

question 13 (5 points) find a logarithmic function to model the data. x y 1 60 2 54 3 51 4 50 5 46 6 45 7 44 f(x)=60.73(0.95)^{x} f(x)=0.93(60.73)^{x} f(x)=60.04 - 8.25 \ln x f(x)=8.25 - 60.04 \ln x

Explanation:

Step1: Recall the form of a logarithmic function

The general form of a logarithmic function is \(y = a + b\ln x\).

Step2: Substitute \(x = 1\) into the function

When \(x = 1\), \(\ln(1)=0\). For the function \(y=a + b\ln x\), when \(x = 1\), \(y=a\). Looking at the data, when \(x = 1\), \(y = 60\).
For option \(f(x)=60.04−8.25\ln x\), when \(x = 1\), \(f(1)=60.04-8.25\ln(1)=60.04\approx60\) (since \(\ln(1) = 0\)).
For option \(f(x)=8.25−60.04\ln x\), when \(x = 1\), \(f(1)=8.25-60.04\ln(1)=8.25
eq60\).
Options \(f(x)=60.73(0.95)^{x}\) and \(f(x)=0.93(60.73)^{x}\) are exponential functions (\(y = ab^{x}\) form), not logarithmic (\(y=a + b\ln x\) form).

Step3: Check another point

Let's check \(x = 2\) for \(f(x)=60.04−8.25\ln x\).
We know that \(\ln(2)\approx0.693\). Then \(f(2)=60.04-8.25\times0.693=60.04 - 5.72725=54.31275\approx54\)

Answer:

\(f(x)=60.04−8.25\ln x\)