A logarithmic function of the form \( y=a+b \log (x+3) \) passes through the points \( (-2,7) \) and \( (31,-4) \). Find \( a \) and \( b \), rounding all values to three decimal places, if necessary. Answer Attempt 1 out of 2 \( a=\square b=\square \)

To find the values of \( a \) and \( b \) in the logarithmic function \( y = a + b \log(x + 3) \) that passes through the points \( (-2, 7) \) and \( (31, -4) \), we can set up two equations based on the given points. 1. For the point \( (-2, 7) \): \[ 7 = a + b \log(-2 + 3) \] Simplifying gives: \[ 7 = a + b \log(1) \] Since \( \log(1) = 0 \), we have: \[ 7 = a \quad \text{(1)} \] 2. For the point \( (31, -4) \): \[ -4 = a + b \log(31 + 3) \] Simplifying gives: \[ -4 = a + b \log(34) \quad \text{(2)} \] Now we will substitute equation (1) into equation (2): \[ -4 = 7 + b \log(34) \] \[ -4 - 7 = b \log(34) \] \[ -11 = b \log(34) \] \[ b = \frac{-11}{\log(34)} \] Now, we calculate \( \log(34) \): \[ \log(34) \approx 1.5315 \] Therefore, \[ b \approx \frac{-11}{1.5315} \approx -7.184 \quad \text{(to three decimal places: -7.184)} \] Using the value of \( b \) in equation (1): \[ a = 7 \] Thus, rounding where necessary, we find: \[ a = 7, \quad b \approx -7.184 \] Final answer: \( a = 7 \quad b \approx -7.184 \)