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Câu hỏi: Cho $a$, $b$ là các số thực dương khác $1$ thỏa mãn ${{\log }_{a}}b=\sqrt{3}$. Giá trị của ${{\log }_{\dfrac{\sqrt{b}}{a}}}\left( \dfrac{\sqrt[3]{b}}{\sqrt{a}} \right)$
A. $-\sqrt{3}$.
B. $-2\sqrt{3}$.
C. $\sqrt{3}$.
D. $-\dfrac{1}{\sqrt{3}}$.
A. $-\sqrt{3}$.
B. $-2\sqrt{3}$.
C. $\sqrt{3}$.
D. $-\dfrac{1}{\sqrt{3}}$.
Ta có ${{\log }_{a}}b=\sqrt{3}\Leftrightarrow b={{a}^{\sqrt{3}}}$.
Khi đó ${{\log }_{\dfrac{\sqrt{b}}{a}}}\left( \dfrac{\sqrt[3]{b}}{\sqrt{a}} \right)={{\log }_{\dfrac{\sqrt{{{a}^{\sqrt{3}}}}}{a}}}\dfrac{\sqrt[3]{{{a}^{\sqrt{3}}}}}{\sqrt{a}}={{\log }_{{{a}^{\dfrac{\sqrt{3}}{2}-1}}}}{{a}^{\dfrac{\sqrt{3}}{3}-\dfrac{1}{2}}}=\left( \dfrac{\sqrt{3}}{3}-\dfrac{1}{2} \right):\left( \dfrac{\sqrt{3}}{2}-1 \right)=-\dfrac{1}{\sqrt{3}}$.
Khi đó ${{\log }_{\dfrac{\sqrt{b}}{a}}}\left( \dfrac{\sqrt[3]{b}}{\sqrt{a}} \right)={{\log }_{\dfrac{\sqrt{{{a}^{\sqrt{3}}}}}{a}}}\dfrac{\sqrt[3]{{{a}^{\sqrt{3}}}}}{\sqrt{a}}={{\log }_{{{a}^{\dfrac{\sqrt{3}}{2}-1}}}}{{a}^{\dfrac{\sqrt{3}}{3}-\dfrac{1}{2}}}=\left( \dfrac{\sqrt{3}}{3}-\dfrac{1}{2} \right):\left( \dfrac{\sqrt{3}}{2}-1 \right)=-\dfrac{1}{\sqrt{3}}$.
Đáp án D.