Solve the following logarithmic equation. \( \log (6 x+3)=1+\log (x-6) \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. (a) A. The solution set is \( \{\mid \) B. There is no solution. (Simplify your answer. Type an exact answer. Use a comma to separate answers as needed. B. The

To solve the logarithmic equation \( \log (6x + 3) = 1 + \log (x - 6) \), we can first isolate the logarithmic expressions. The equation can be rewritten using the property of logarithms that states \( \log a + \log b = \log (ab) \). Starting from the equation: \[ \log (6x + 3) = \log (10) + \log (x - 6) \] This simplifies to: \[ \log (6x + 3) = \log (10(x - 6)) \] Now, since the logarithms are equal, we can set the insides equal to each other: \[ 6x + 3 = 10(x - 6) \] Expanding the right side gives: \[ 6x + 3 = 10x - 60 \] Next, we rearrange this equation: \[ 3 + 60 = 10x - 6x \] This simplifies to: \[ 63 = 4x \] Dividing both sides by 4: \[ x = \frac{63}{4} = 15.75 \] Now, we need to check that this value is valid in the original logarithmic functions. The arguments \( 6x + 3 \) and \( x - 6 \) must both be positive: 1. For \( 6(15.75) + 3 = 95.5 > 0 \), so this is valid. 2. For \( 15.75 - 6 = 9.75 > 0 \), so this is also valid. Thus, the solution set is: \( \{ 15.75 \} \) So the correct response is: (A) The solution set is \( \{ 15.75 \} \).