Cho {log _a}b = sqrt 3 .) Tình ({log _{frac{{sqrt b }}{a}}}frac{{sqrt b }}{{sqrt a }}.

Ta có:

 \(\begin{array}{l}{\log _{\frac{{\sqrt b }}{a}}}\frac{{\sqrt b }}{{\sqrt a }} = {\log _{\frac{{\sqrt b }}{a}}}\sqrt b  - {\log _{\frac{{\sqrt b }}{a}}}\sqrt a  = \frac{1}{{{{\log }_{\sqrt b }}\left( {\frac{{\sqrt b }}{a}} \right)}} - \frac{1}{{{{\log }_{\sqrt a }}\left( {\frac{{\sqrt b }}{a}} \right)}}\\ = \frac{1}{{2\left( {{{\log }_b}\sqrt b  - {{\log }_b}a} \right)}} - \frac{1}{{2\left( {{{\log }_a}\sqrt b  - {{\log }_a}a} \right)}} = \frac{1}{{2\left( {\frac{1}{2} - \frac{1}{{{{\log }_a}b}}} \right)}} - \frac{1}{{2\left( {\frac{1}{2}{{\log }_a}b - 1} \right)}}\end{array}\)

\( = \frac{1}{{2\left( {\frac{1}{2} - \frac{1}{{\sqrt 3 }}} \right)}} - \frac{1}{{2\left( {\frac{{\sqrt 3 }}{2} - 1} \right)}} = \frac{{\sqrt 3  - 1}}{{\sqrt 3  - 2}}.\)