Solve the equation. \[ e^{3 x-1}=\left(e^{6}\right)^{-x} \] A. $\left\{-\frac{1}{3}\right\}$ B. $\left\{\frac{7}{4}\right\}$ C. $\left\{\frac{1}{9}\right\}$ D. $\{0\}$

Step 1 ：The given equation is in the form of exponential functions. We can simplify the equation by taking the natural logarithm on both sides. This will allow us to solve for x.

Step 2 ：Solving the equation, we get multiple solutions. However, some of these solutions include complex numbers, which are not in the options. Therefore, we only consider the real solution.

Step 3 ：The real solution to the equation is \(\frac{1}{9}\).

Step 4 ：Final Answer: \(\boxed{\left\{\frac{1}{9}\right\}}\)