A logarithmic function of the form \( y=a+b \log (x+4) \) passes through the points \( (-3,-2) \) and \( (773,10) \). 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 \( y = a + b \log(x + 4) \) that passes through the points \( (-3, -2) \) and \( (773, 10) \), we can set up a system of equations based on these points. 1. **Substituting the first point \((-3, -2)\)**: \[ -2 = a + b \log(-3 + 4) \] Simplifying this gives: \[ -2 = a + b \log(1) \] Since \( \log(1) = 0 \), we have: \[ -2 = a \] 2. **Substituting the second point \((773, 10)\)**: \[ 10 = a + b \log(773 + 4) \] Simplifying this gives: \[ 10 = a + b \log(777) \] Now we can substitute \( a = -2 \) into the second equation: \[ 10 = -2 + b \log(777) \] This simplifies to: \[ 12 = b \log(777) \] Now we can solve for \( b \): \[ b = \frac{12}{\log(777)} \] Next, we will calculate \( b \) using the logarithm. Let's perform the calculations to find \( b \) and confirm \( a \). Numbers by following steps: - step0: Solution: \(-2\) Calculate or simplify the expression \( 12 / \log(777) \). Calculate the value by following steps: - step0: Calculate: \(\frac{12}{\log_{10}{\left(777\right)}}\) After performing the calculations, we find: - \( a = -2 \) - \( b \approx 4.152 \) (rounded to three decimal places) Thus, the final values are: \[ a = -2, \quad b \approx 4.152 \]