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. \[ \log _{3} x+\log _{3}(2 x-1)=1 \]

Step 1 ：The given equation is \(\log _{3} x+\log _{3}(2 x-1)=1\).

Step 2 ：We can use the properties of logarithms to simplify the equation. The property \(\log_b(m) + \log_b(n) = \log_b(mn)\) allows us to combine the two logarithmic expressions on the left side of the equation. This gives us \(\log _{3}(x(2x-1))=1\).

Step 3 ：We can convert the logarithmic equation to an exponential equation to solve for x. This gives us \(3 = x(2x - 1)\).

Step 4 ：Solving this equation gives us two solutions, -1 and 3/2.

Step 5 ：However, only 3/2 is a valid solution because it is the only one that falls within the domain of the original logarithmic expressions. The domain of a logarithmic function is (0, ∞), so any solution that is less than or equal to 0 is not valid.

Step 6 ：Final Answer: The solution to the logarithmic equation is \(\boxed{\frac{3}{2}}\).