Solve the logarithmic equation. Be sure to reject any value of $x$ that is not in the domain of the original logarithmic expressions. Give an exact answer.
\[
\log _{13} x+\log _{13}(12 x-1)=1
\]
Rewrite the given equation without logarithms. Do not solve for $\mathrm{x}$.
Solve the equation. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is
(Type an exact answer in simplified form. Use integers or fractions for any numbers in the expression.)
B. There are infinitely many solutions.
C. There is no solution.

Step 1 ：Given the logarithmic equation \(\log _{13} x+\log _{13}(12 x-1)=1\).

Step 2 ：According to the logarithmic properties, this can be rewritten as \(\log _{13} (x \cdot (12x - 1))=1\).

Step 3 ：Then, we can convert the logarithmic equation to an exponential equation. The base is 13, the exponent is 1, and the result is \(x \cdot (12x - 1)\). So, the equation becomes \(13^1 = x \cdot (12x - 1)\).

Step 4 ：Solving this equation, we get the solutions as \(x = -1\) and \(x = \frac{13}{12}\).

Step 5 ：However, we need to check if these solutions are in the domain of the original logarithmic expressions. The domain of a logarithmic function is \((0, \infty)\). Therefore, \(x = -1\) is not in the domain of the original logarithmic expressions and must be rejected.

Step 6 ：The only solution is \(x = \frac{13}{12}\).

Step 7 ：Final Answer: The solution set is \(\boxed{\frac{13}{12}}\).