Solve the equation. Write the solution set with the exact solutions. \[ \ln x+\ln (x-4)=\ln (3 x-10) \] If there is more than one solution, separate the answers with commas

Step 1 ：The given equation is a logarithmic equation. To solve it, we can use the properties of logarithms to simplify the equation and then solve for x. The properties of logarithms we will use are: \(\ln a + \ln b = \ln (ab)\) and if \(\ln a = \ln b\), then \(a = b\).

Step 2 ：We can apply the first property to combine the left side of the equation. So, \(\ln x+\ln (x-4)=\ln (x(x-4))\).

Step 3 ：Then, we can apply the second property to equate the arguments of the logarithms. So, \(x(x - 4) = 3x - 10\).

Step 4 ：Solving the resulting equation for x, we get the solutions x = 2 and x = 5.

Step 5 ：However, we need to check these solutions in the original equation because the domain of the logarithmic function is (0, ∞). Therefore, x must be greater than 0 and (x-4) must also be greater than 0.

Step 6 ：After checking, we find that only x = 5 is a valid solution.

Step 7 ：Final Answer: The valid solution to the equation is \(\boxed{5}\).