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

Hàm số mũ | Hàm số lũy thừa | Hàm số mũ và lũy thừa | hàm số loagrit | logarit |
Cho \({\log _a}b = \sqrt 3 .\) Tình \({\log _{\frac{{\sqrt b }}{a}}}\frac{{\sqrt b }}{{\sqrt a }}.\)
A. \(\frac{{\sqrt 3 - 1}}{{\sqrt 3 - 2}}\)
B. \(\sqrt 3 + 1\)
C. \(\frac{{\sqrt 3 - 1}}{{\sqrt 3 + 2}}\)
D. \(\sqrt 3 - 1\)

Ta có:
\(\begin{array}{l}{\log _{\frac{{\sqrt b }}{a}}}\frac{{\sqrt b }}{{\sqrt a }} = {\log _{\frac{{\sqrt b }}{a}}}\sqrt b - {\log _{\frac{{\sqrt b }}{a}}}\sqrt a = \frac{1}{{{{\log }_{\sqrt b }}\left( {\frac{{\sqrt b }}{a}} \right)}} - \frac{1}{{{{\log }_{\sqrt a }}\left( {\frac{{\sqrt b }}{a}} \right)}}\\ = \frac{1}{{2\left( {{{\log }_b}\sqrt b - {{\log }_b}a} \right)}} - \frac{1}{{2\left( {{{\log }_a}\sqrt b - {{\log }_a}a} \right)}} = \frac{1}{{2\left( {\frac{1}{2} - \frac{1}{{{{\log }_a}b}}} \right)}} - \frac{1}{{2\left( {\frac{1}{2}{{\log }_a}b - 1} \right)}}\end{array}\)
\( = \frac{1}{{2\left( {\frac{1}{2} - \frac{1}{{\sqrt 3 }}} \right)}} - \frac{1}{{2\left( {\frac{{\sqrt 3 }}{2} - 1} \right)}} = \frac{{\sqrt 3 - 1}}{{\sqrt 3 - 2}}.\)