A logarithmic function of the form \( y=a+b \log (x-4) \) passes through the points \( (5,17) \) and \( (879,-38) \). Find \( a \) and \( b \), rounding all values to three decimal places, if necessary. Answer Attempt 1 out of 2 \( a=\square b=\square \)

We are given the function \[ y = a + b\,\log(x-4) \] and the points \((5, 17)\) and \((879, -38)\). **Step 1. Find \(a\) using the point \((5,17)\).** Substitute \(x=5\) and \(y=17\) into the equation: \[ 17 = a + b\,\log(5-4) \] Calculate the argument of the logarithm: \[ 5-4 = 1 \] Since \(\log(1)=0\) (regardless of the base), \[ 17 = a + b\cdot0 \quad\Rightarrow\quad a = 17. \] **Step 2. Find \(b\) using the point \((879,-38)\).** Substitute \(x=879\) and \(y=-38\) into the equation: \[ -38 = 17 + b\,\log(879-4). \] Calculate the argument of the logarithm: \[ 879-4 = 875. \] Now we have: \[ -38 = 17 + b\,\log(875). \] Subtract \(17\) from both sides: \[ -38 - 17 = b\,\log(875) \quad\Rightarrow\quad -55 = b\,\log(875). \] Now solve for \(b\): \[ b = \frac{-55}{\log(875)}. \] Assuming that the logarithm \(\log\) is the common logarithm (base 10), we need to compute \(\log_{10}(875)\). **Step 3. Calculate \(\log_{10}(875)\) and Numerical Value of \(b\).** Write 875 as \[ 875 = 8.75 \times 10^2, \] so \[ \log_{10}(875)=\log_{10}(8.75) + \log_{10}(10^2)=\log_{10}(8.75)+2. \] Using a calculator, we find \[ \log_{10}(8.75) \approx 0.942, \] so \[ \log_{10}(875) \approx 0.942 + 2 = 2.942. \] Now substitute back to find \(b\): \[ b=\frac{-55}{2.942}\approx -18.699 \quad (\text{rounded to three decimal places}). \] **Final Answer:** \[ a=17.000\quad \text{and} \quad b=-18.699. \]