Cho \({\log _{\dfrac{1}{2}}}\left( {\dfrac{1}{5}} \right) = a\). Khẳng định nào dưới đây đúng?

* Hướng dẫn giải

Ta có :

\({\log _{\dfrac{1}{2}}}\left( {\dfrac{1}{5}} \right) = a \Leftrightarrow {\log _{{2^{ - 1}}}}\left( {{5^{ - 1}}} \right) = a\) \( \Leftrightarrow \dfrac{1}{{\left( { - 1} \right)}}.\left( { - 1} \right).{\log _2}5 = a \Leftrightarrow a = {\log _2}5\)

Suy ra :

\({\log _2}25 + {\log _2}\sqrt 5  = {\log _2}{5^2} + {\log _2}{5^{\dfrac{1}{2}}}\) \( = 2{\log _2}5 + \dfrac{1}{2}{\log _2}5 = \dfrac{5}{2}{\log _2}5 = \dfrac{5}{2}a\)

\({\log _5}4 = \dfrac{1}{{{{\log }_4}5}} = \dfrac{1}{{{{\log }_{{2^2}}}5}} = \dfrac{1}{{\dfrac{1}{2}{{\log }_2}5}} = \dfrac{2}{a}\)

\({\log _2}\dfrac{1}{5} + {\log _2}\dfrac{1}{{25}} = {\log _2}{5^{ - 1}} + {\log _2}{5^{ - 2}}\) \( = \left( { - 3} \right).{\log _2}5 =  - 3a\)

Chọn B