$\ln x+\ln (x-2)=\ln (6 x)$, then $x=$ Question Help: $\square$ Video $\square$ Message instructor Submit Question Jump to Answer

Step 1 ：The given equation is in the form of logarithms: \(\ln x+\ln (x-2)=\ln (6 x)\).

Step 2 ：We can use the properties of logarithms to simplify the equation and solve for x. The property we will use is that the sum of logarithms is equal to the logarithm of the product. So, \(\ln a + \ln b = \ln (ab)\).

Step 3 ：We can apply this property to the left side of the equation to simplify it: \(\ln (x*(x - 2))\).

Step 4 ：Then, we can equate the simplified left side to the right side: \(\ln (x*(x - 2)) = \ln (6*x)\).

Step 5 ：Solving for x, we find that the solution to the equation is x = 8. This means that when we substitute x = 8 into the original equation, both sides of the equation will be equal, thus satisfying the equation.

Step 6 ：Final Answer: The solution to the equation is \(\boxed{8}\).