Solve the logarithmic equation. Be sure to reject any value of $x$ that is not in the domain of the original logarithmic expressions. Give the exact answer.
\[
\log (x+14)-\log 2=\log (5 x+6)
\]
Solve the equation to find the solution set. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is
(Simplify your answer. Use a comma to separate answers as needed.)
B. There are infinitely many solutions.
C. There is no solution.

Step 1 ：The given equation is a logarithmic equation. The first step to solve this equation is to combine the logarithms on the left side of the equation using the properties of logarithms. The property that states that the difference of two logarithms is the logarithm of the quotient of their arguments can be used here. After combining the logarithms, the equation can be rewritten as a single logarithm on both sides. Then, the property that states that if two logarithms with the same base are equal, their arguments are also equal can be used to solve for x.

Step 2 ：The solution to the equation is \(x = \frac{2}{9}\). However, we need to check if this solution is in the domain of the original logarithmic expressions. The domain of a logarithmic function is the set of all real numbers for which the argument of the logarithm is greater than zero. Therefore, we need to check if \(x+14 > 0\) and \(5x+6 > 0\) for \(x = \frac{2}{9}\).

Step 3 ：The solution \(x = \frac{2}{9}\) is in the domain of the original logarithmic expressions since both \(x+14 > 0\) and \(5x+6 > 0\) are true for \(x = \frac{2}{9}\). Therefore, the solution to the equation is valid.

Step 4 ：Final Answer: The solution set is \(\boxed{\frac{2}{9}}\).