Cho 2 số thực dương \(a, b\) thỏa mãn \(\sqrt a  \ne b,a \ne 1,{\log _a}b = 2\).

* Hướng dẫn giải

Ta có \({\log _a}b = 2 \Rightarrow {\log _b}a = \frac{1}{2}\)

\(\begin{array}{l}
T = {\log _{\frac{{\sqrt a }}{b}}}\sqrt[3]{{ba}} = {\log _{\frac{{\sqrt a }}{b}}}\sqrt[3]{b} + {\log _{\frac{{\sqrt a }}{b}}}\sqrt[3]{a} = \frac{1}{{{{\log }_{\sqrt[3]{b}}}\frac{{\sqrt a }}{b}}} + \frac{1}{{{{\log }_{\sqrt[3]{a}}}\frac{{\sqrt a }}{b}}}\\
 = \frac{1}{{{{\log }_{\sqrt[3]{b}}}\sqrt a  - {{\log }_{\sqrt[3]{b}}}b}} + \frac{1}{{{{\log }_{\sqrt[3]{a}}}\sqrt a  - {{\log }_{\sqrt[3]{a}}}b}}\\
 = \frac{1}{{\frac{3}{2}{{\log }_b}a - 3}} + \frac{1}{{\frac{3}{2} - 3{{\log }_a}b}} = \frac{1}{{\frac{3}{2}.\frac{1}{2} - 3}} + \frac{1}{{\frac{3}{2} - 3.2}} =  - \frac{2}{3}
\end{array}\)