Solve the logarithmic equation. Express all solutions in exact form. \[ \log _{3}\left(x^{3}-37\right)=3 \] Select the correct choice below and fill in any answer boxes in your choice. A. The solution set is \{\} . (Simplify your answer. Use a comma to separate answers as needed.) B. The solution is the empty set.

Step 1 ：Given the logarithmic equation \(\log _{3}\left(x^{3}-37\right)=3\).

Step 2 ：To solve for x, we need to convert the logarithmic equation into an exponential equation. The base of the logarithm is 3 and the exponent is 3. So, we can rewrite the equation as \(3^3 = x^3 - 37\).

Step 3 ：Solving for x, we get three solutions: \(4\), \(-2 - 2\sqrt{3}i\), and \(-2 + 2\sqrt{3}i\).

Step 4 ：However, since we are dealing with a logarithmic equation, we need to discard any solution that would make the argument of the logarithm negative or zero, as the logarithm is undefined for these values.

Step 5 ：Looking at the solutions, we can see that the first one, 4, is the only real number and would make the argument of the logarithm positive. Therefore, this is the only valid solution.

Step 6 ：Final Answer: The solution to the logarithmic equation is \(\boxed{4}\).