Question

Tính giá trị biểu thức :

              \(A=\log_{\frac{\sqrt{b}}{a}}\frac{\sqrt[3]{b}}{\sqrt{a}}\) biết \(\log_ab=\sqrt{3}\)

Answer 1

\(A=\log_{\frac{\sqrt{b}}{a}}\frac{\sqrt[3]{b}}{\sqrt{a}}=\log_{\frac{\sqrt{b}}{a}}b^{\frac{1}{3}}-\log_{\frac{\sqrt{b}}{a}}a^{\frac{1}{3}}=\frac{1}{3\log_b\frac{\sqrt{b}}{a}}-\frac{1}{2\log_a\frac{\sqrt{b}}{a}}\)

    \(=\frac{1}{3\left(\frac{1}{2}-\log_ba\right)}-\frac{1}{2\left(\frac{1}{2}\log_ab-1\right)}\)

    \(=\frac{1}{3\left(\frac{1}{2}-\log_ba\right)}-\frac{1}{\log_ab-2}=\frac{a\log_ab}{3\left(\log_ab-2\right)}-\frac{1}{\log_ab-2}\)

   \(=\frac{2\sqrt{3}-3}{3\left(\sqrt{3}-2\right)}=-\frac{\sqrt{3}}{3}\)