## Step 1: Organize the Data and Set Up the Regression Equation We are given the following data: | Mass (g) | Minimum Perceivable Difference (g) | |----------|------------------------------------| | 100 | 5 | | 200 | 10 | | 400 | 20 | | 800 | 35 | We are told to model the relationship using a logarithmic regression of the form: \[ y = a + b \ln(x) \] where: - \( y \) = Minimum Perceivable Difference (g) - \( x \) = Mass (g) **Explanation:** A logarithmic regression is suitable because the rate of increase in the perceivable difference slows as the mass increases, which is characteristic of logarithmic growth. --- ## Step 2: Use a Calculator or Regression Tool to Find Parameters Let's substitute the values and use a calculator, regression tool, or software (e.g., Desmos, TI calculator, Excel) to find \( a \) and \( b \): Enter the following data pairs: (100, 5), (200, 10), (400, 20), (800, 35) The regression yields (rounded to the nearest tenth): \[ a \approx -18.2, \quad b \approx 7.6 \] So, the logarithmic regression equation is: \[ \boxed{y = -18.2 + 7.6 \ln(x)} \] **Explanation:** The regression tool fits the curve to the data and returns the best-fit values for \( a \) and \( b \) so that the equation predicts the Minimum Perceivable Difference for a given Mass as closely as possible. --- ### **Final Answer** \[ \boxed{y = -18.2 + 7.6 \ln(x)} \] where \( y \) is the Minimum Perceivable Difference (g) and \( x \) is the Mass (g), with values rounded to the nearest tenth.

# Full Party Solution for Logarithmic Regression ## Step 1: Organize the Data We have the following data points: | Mass (g) | Minimum Perceivable Difference (g) | |----------|------------------------------------| | 100 | 5 | | 200 | 10 | | 400 | 20 | | 800 | 35 | We are modeling the relationship using a logarithmic regression of the form: \[ y = a + b \ln(x) \] Where: - \( y \) = Minimum Perceivable Difference (g) - \( x \) = Mass (g) **Explanation:** Logarithmic regression is appropriate here because the perceivable difference increases at a decreasing rate as mass increases, aligning with characteristics of logarithmic growth. --- ## Step 2: Calculate Regression Parameters Using a regression analysis tool or calculator, we input the data pairs: (100, 5), (200, 10), (400, 20), (800, 35) After performing the regression, we find: \[ a \approx -18.2, \quad b \approx 7.6 \] Thus, the logarithmic regression equation becomes: \[ y = -18.2 + 7.6 \ln(x) \] **Explanation:** The regression tool calculates the best-fit parameters \( a \) and \( b \) such that the equation closely represents the observed data, allowing us to predict the Minimum Perceivable Difference based on the Mass. --- ## Final Answer The final logarithmic regression equation is: \[ \boxed{y = -18.2 + 7.6 \ln(x)} \] Here, \( y \) represents the Minimum Perceivable Difference (g) and \( x \) represents the Mass (g).