91Ó°ÊÓ

Given the expression: \(2(\ln x)-3\left(\ln x^{2}+\ln x\right)\), we can simplify and rewrite it as a single logarithm by following these steps: Step 1: Distribute the coefficients to the logarithms within the parentheses: $$2(\ln x)-3\left(\ln x^{2}+\ln x\right) = 2\ln x - 3\ln x^{2} - 3\ln x$$ Step 2: Apply the power property of logarithms to the expression: $$ 2\ln x - 3\ln x^{2} - 3\ln x = 2\ln x - 6\ln x - 3\ln x$$ Step 3: Combine using the product and quotient properties of logarithms: $$2\ln x - 6\ln x - 3\ln x = \ln x^{2} - \ln x^{6} - \ln x^3 = \ln \left(\frac{x^2}{x^6} \cdot\frac{1}{x^3} \right)$$ Step 4: Simplify the argument of the logarithm: $$\ln \left(\frac{x^2}{x^6} \cdot\frac{1}{x^3}\right) = \ln \left (\frac{x^2}{x^6x^3}\right) = \ln \left (\frac{x^2}{x^9}\right)$$ Step 5: Apply the quotient property of logarithms once more: $$\ln \left (\frac{x^2}{x^9}\right) = \ln x^2 - \ln x^9$$ Therefore, the given expression can be simplified and rewritten as a single logarithm: \(\ln x^2 - \ln x^9\).