Cho hai số thực dương a, b thỏa mãn {log _a}b = 2. Tính {log _{frac{{sqrt a }}{b}}}left( {sqrt[3]{b}a} ight).

$\begin{array}{l}{\log _{\frac{{\sqrt a }}{b}}}\left( {\sqrt[3]{b}.a} \right) = \frac{1}{3}{\log _{\frac{{\sqrt a }}{b}}}b + {\log _{\frac{{\sqrt a }}{b}}}a = \frac{1}{3}.\frac{1}{{{{\log }_b}\left( {\frac{{\sqrt a }}{b}} \right)}} + \frac{1}{{{{\log }_a}\left( {\frac{{\sqrt a }}{b}} \right)}}\\ = \frac{1}{{3\left( {{{\log }_b}\sqrt a  - {{\log }_b}b} \right)}} + \frac{1}{{{{\log }_a}\sqrt a  - {{\log }_a}b}}\end{array}$

$= \frac{1}{{3\left( {\frac{1}{2}{{\log }_b}a - 1} \right)}} + \frac{1}{{\frac{1}{2} - {{\log }_a}b}} = \frac{1}{{3\left( {\frac{1}{4} - 1} \right)}} + \frac{1}{{\frac{1}{2} - 2}} =  - \frac{{10}}{9}.$