Biết

Câu hỏi: Biết $\log_{a} b = 2$ . Tính $\log_{\frac{b}{a}} \frac{\sqrt[3]{a^{2}}}{\sqrt{b^{3}}}$ .

A. $- \frac{12}{5}$ .

B. $\frac{11}{3}$ .

C. $- \frac{7}{3}$ .

D. $\frac{13}{6}$ .

Đáp án

- Hướng dẫn giải

Đáp án C

$\begin{array}{l} \log_{\frac{b}{a}} \frac{\sqrt[3]{a^{2}}}{\sqrt{b^{3}}} = \log_{\frac{b}{a}} a^{2 / 3} - \log_{\frac{b}{a}} b^{3 / 2} \\ = \frac{2}{3} \log_{\frac{b}{a}} a - \frac{3}{2} \log_{\frac{b}{a}} b \\ = \frac{2}{3} . \frac{1}{\log_{a} (\frac{b}{a})} - \frac{3}{2} . \frac{1}{\log_{b} (\frac{b}{a})} \\ = \frac{2}{3} . \frac{1}{\log_{a} b - 1} - \frac{3}{2} . \frac{1}{1 - \log_{b} a} \\ = \frac{2}{3} . \frac{1}{2 - 1} - \frac{3}{2} . \frac{1}{1 - \frac{1}{2}} = - \frac{7}{3} . \end{array}$