A logarithmic function of the form \( y=a+b \ln (x) \) passes through the points \( (1,-17) \) and \( (690,17) \). 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 \( y = a + b \ln(x) \), we can use the two given points \( (1, -17) \) and \( (690, 17) \). Plugging in the first point \( (1, -17) \): \[ -17 = a + b \ln(1) \] Since \( \ln(1) = 0 \), we simplify this to: \[ -17 = a \quad \Rightarrow \quad a = -17. \] Now, using the second point \( (690, 17) \): \[ 17 = -17 + b \ln(690). \] Solving for \( b \): \[ 17 + 17 = b \ln(690) \quad \Rightarrow \quad 34 = b \ln(690). \] Now, we compute \( \ln(690) \): \[ \ln(690) \approx 6.531. \] Substituting this into the equation gives: \[ 34 = b \cdot 6.531 \quad \Rightarrow \quad b = \frac{34}{6.531} \approx 5.206. \] Rounding the values, we have: \[ a \approx -17, \quad b \approx 5.206. \] Final answer: \( a = -17, \quad b \approx 5.206 \)