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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 $\log _{\dfrac{\sqrt{b}}{}}\left(\dfrac{\sqrt[3]{b}}{\sqrt{a}}\right)$ là
A. $-2 \sqrt{3}$.
B. $-\sqrt{3}$.
C. $\sqrt{3}$.
D. $-\dfrac{1}{\sqrt{3}}$.
A. $-2 \sqrt{3}$.
B. $-\sqrt{3}$.
C. $\sqrt{3}$.
D. $-\dfrac{1}{\sqrt{3}}$.
$
\begin{aligned}
& \text { Ta có: } \log _{\dfrac{\sqrt{b}}{}}^a\left(\dfrac{\sqrt[3]{b}}{\sqrt{a}}\right)=\dfrac{\log _a\left(\dfrac{\sqrt[3]{b}}{\sqrt{a}}\right)}{\log _a\left(\dfrac{\sqrt{b}}{a}\right)}=\dfrac{\log _a \sqrt[3]{b}-\log _a \sqrt{a}}{\log _a \sqrt{b}-\log _a a} . \\
& =\dfrac{\dfrac{1}{3} \log _a b-\dfrac{1}{2} \log _a a}{\dfrac{1}{2} \log _a b-\log _a a}=\dfrac{\dfrac{1}{3} \sqrt{3}-\dfrac{1}{2}}{\dfrac{1}{2} \sqrt{3}-1}=-\dfrac{1}{\sqrt{3}} .
\end{aligned}
$
\begin{aligned}
& \text { Ta có: } \log _{\dfrac{\sqrt{b}}{}}^a\left(\dfrac{\sqrt[3]{b}}{\sqrt{a}}\right)=\dfrac{\log _a\left(\dfrac{\sqrt[3]{b}}{\sqrt{a}}\right)}{\log _a\left(\dfrac{\sqrt{b}}{a}\right)}=\dfrac{\log _a \sqrt[3]{b}-\log _a \sqrt{a}}{\log _a \sqrt{b}-\log _a a} . \\
& =\dfrac{\dfrac{1}{3} \log _a b-\dfrac{1}{2} \log _a a}{\dfrac{1}{2} \log _a b-\log _a a}=\dfrac{\dfrac{1}{3} \sqrt{3}-\dfrac{1}{2}}{\dfrac{1}{2} \sqrt{3}-1}=-\dfrac{1}{\sqrt{3}} .
\end{aligned}
$
Đáp án D.