Google
Câu hỏi: Cho $a$, $b$ là các số thực dương khác $1$ thỏa mãn ${{\log }_{a}}b=\sqrt{3}$. Giá trị của ${{\log }_{\dfrac{\sqrt{b}}{a}}}\left( \dfrac{\sqrt[3]{b}}{\sqrt{a}} \right)$ là:
A. $-\sqrt{3}$.
B. $-\dfrac{1}{\sqrt{3}}$.
C. $-2\sqrt{3}$.
D. $\sqrt{3}$.
A. $-\sqrt{3}$.
B. $-\dfrac{1}{\sqrt{3}}$.
C. $-2\sqrt{3}$.
D. $\sqrt{3}$.
${{\log }_{a}}b=\sqrt{3}$ $\Rightarrow b={{a}^{\sqrt{3}}}$.
${{\log }_{\dfrac{\sqrt{b}}{a}}}\left( \dfrac{\sqrt[3]{b}}{\sqrt{a}} \right)$ $={{\log }_{{{a}^{\left( \dfrac{\sqrt{3}}{2}-1 \right)}}}}\left( {{a}^{\left( \dfrac{\sqrt{3}}{3}-\dfrac{1}{2} \right)}} \right)$ $=\dfrac{\left( 2\sqrt{3}-3 \right)2}{6\left( \sqrt{3}-2 \right)}$ $=-\dfrac{1}{\sqrt{3}}$.
${{\log }_{\dfrac{\sqrt{b}}{a}}}\left( \dfrac{\sqrt[3]{b}}{\sqrt{a}} \right)$ $={{\log }_{{{a}^{\left( \dfrac{\sqrt{3}}{2}-1 \right)}}}}\left( {{a}^{\left( \dfrac{\sqrt{3}}{3}-\dfrac{1}{2} \right)}} \right)$ $=\dfrac{\left( 2\sqrt{3}-3 \right)2}{6\left( \sqrt{3}-2 \right)}$ $=-\dfrac{1}{\sqrt{3}}$.
Đáp án B.