Questions: Student Homework 3.5: Exponential and Logarithmic Models Score: 18.5 / 25 10 / 13 answered Question 12 x 1 2 3 4 5 6 y 726 960 1160 1663 2116 2772 Use regression to find an exponential equation that best fits the data above. The equation has form y = a * b^x where: a =□ b =□

Student

Homework 3.5: Exponential and Logarithmic Models
Score: 18.5 / 25 10 / 13 answered
Question 12

x 1 2 3 4 5 6
y 726 960 1160 1663 2116 2772

Use regression to find an exponential equation that best fits the data above. The equation has form y = a * b^x where:

a =□
b =□

Transcript text: KeMyTcC: Student Homework 3.5: Exponential and Logarithmic Models Score: $18.5 / 25 \quad 10 / 13$ answered Question 12 \begin{tabular}{|r|r|r|r|r|r|r|} \hline $\mathbf{x}$ & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline $\mathbf{y}$ & 726 & 960 & 1160 & 1663 & 2116 & 2772 \\ \hline \end{tabular} Use regression to find an exponential equation that best fits the data above. The equation has form $y=a b^{x}$ where: \[ \begin{array}{l} \mathrm{a}=\square \\ \mathrm{b}=\square \end{array} \] Question Help: Video Message instructor Calculator Submit Question

Solution

Solution Steps

Step 1: Transform the exponential equation to a linear form

By taking the natural logarithm on both sides of the equation $y = a b^{x}$, we get $\ln(y) = \ln(a) + x\ln(b)$. This transformation allows us to use linear regression to find the coefficients $\ln(a)$ and $\ln(b)$.

Step 2: Perform linear regression

Using the transformed data points $(x, \ln(y))$, linear regression yields the coefficients $\ln(a) = 6.309$ and $\ln(b) = 0.269$.

Step 3: Calculate $a$ and $b$

After obtaining the coefficients from the linear regression, we calculate $a = e^{\ln(a)} = 549.456$ and $b = e^{\ln(b)} = 1.309$.