If $\ln x+\ln (x-7)=\ln (3 x)$, then $x=$
Next Question
Final Answer: The solution to the equation is \(\boxed{10}\).
Step 1 :Given the equation in logarithmic form: \(\ln x+\ln (x-7)=\ln (3 x)\)
Step 2 :Using the properties of logarithms, the sum of the logarithms of two numbers is equal to the logarithm of the product of those numbers.
Step 3 :So, we can rewrite the left side of the equation as \(\ln(x*(x-7))\).
Step 4 :Then, we equate it to the right side of the equation and solve for x.
Step 5 :By solving the equation, we find that x = 10.
Step 6 :Substituting x = 10 into the equation, we find that the equation holds true.
Step 7 :Final Answer: The solution to the equation is \(\boxed{10}\).