Solve the logarithmic equation. \[ 2 \ln (x-5)=\ln (x+7)+\ln 6 \] What is the equivalent algebraic equation that must be solved? A. $2(x-5)=6(x+7)$ B. $(x-5)^{2}=6(x+7)$ c. $2(x-5)=(x+7)+6$ D. $(x-5)^{2}=(x+7)+6$ Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $x=$ (Simplify your answer. Use a comma to separate answers as needed.) B. There is no solution.

Step 1 ：Apply the properties of logarithms to the given equation.

Step 2 ：Combine the two logarithms on the right side of the equation using the property \(\ln(a) + \ln(b) = \ln(ab)\). This gives us: \[2 \ln (x-5)=\ln ((x+7) \cdot 6)\]

Step 3 ：Rewrite the left side of the equation using the property \(n \ln(a) = \ln(a^n)\). This gives us: \[\ln ((x-5)^2)=\ln ((x+7) \cdot 6)\]

Step 4 ：Equate the arguments of the logarithms on both sides to get the equivalent algebraic equation: \[(x-5)^2=(x+7) \cdot 6\]

Step 5 ：The equivalent algebraic equation that must be solved is \(\boxed{(x-5)^{2}=6(x+7)}\)