Question 13 If \( \ln x+\ln (x-2)=\ln (4 x) \), then \( x=\square \) Question Help: \( \square \) Video Submit Question

To solve the equation \( \ln x + \ln (x-2) = \ln (4x) \), we can utilize the properties of logarithms. Combining the left side using the property \( \ln a + \ln b = \ln (a \cdot b) \), we rewrite it as \( \ln (x(x-2)) = \ln (4x) \). This leads us to \( x(x-2) = 4x \). Rearranging gives: \( x^2 - 2x - 4x = 0 \) or \( x^2 - 6x = 0 \). Factoring out \( x \), we find \( x(x - 6) = 0 \). Thus \( x = 0 \) or \( x = 6 \). Since the logarithm's argument must be positive, we reject \( x = 0 \). Therefore, the solution is \( x = 6 \)! Isn't it fascinating how logarithmic properties help simplify complex problems into manageable pieces? If you want to dive deeper, check out introductory calculus or algebra materials that cover logarithmic identities and equations. It can empower you to tackle not just this problem but many more in the realm of mathematics!