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