Cho \({\log _{ab}}b = 3\) (với \(a > 0,b > 0,ab \ne 1\)). Tính \({\log _{\sqrt {ab} }}\left( {\frac{a}{{{b^2}}}} \right)\).

Đáp án D

Ta có: \({\log _{ab}}b = 3 \Leftrightarrow {\log _a}ab = \frac{1}{3} \Leftrightarrow {\log _b}a + 1 = \frac{1}{3} \Leftrightarrow {\log _b}a = - \frac{2}{3}\).

Ta có: \({\log _{\sqrt {ab} }}\left( {\frac{a}{{{b^2}}}} \right) = 2{\log _{ab}}\left( {\frac{a}{{{b^2}}}} \right) = 2{\log _{ab}}a - 4{\log _{ab}}b = \frac{2}{{{{\log }_a}ab}} - 4.3 = \frac{2} - 12 = \frac{2}} - 12 = - 16\).