Đề bài

Cho \(a > 0,{\rm{ }}a \ne 1\) và \({a^{\frac{1}{2}}} = b.\) Tính: a) \({\log _a}b;\)                                       b) \({\log _a}\left( {{a^3}{b^2}} \right);\) c) \({\log _{\sqrt a }}\left( {\frac{a}{b}} \right);\)                                d) \({\log _{ab}}\left( {a\sqrt b } \right).\)

Lời giải chi tiết

a) \({\log _a}b = {\log _a}{a^{\frac{1}{2}}} = \frac{1}{2}.\)b) \({\log _a}\left( {{a^3}{b^2}} \right) = {\log _a}\left( {{a^3}.a} \right) = {\log _a}{a^4} = 4.\)c) \({\log _{\sqrt a }}\left( {\frac{a}{b}} \right) = {\log _{\sqrt a }}\left( {\frac{a}{{{a^{\frac{1}{2}}}}}} \right) = 2{\log _a}{a^{\frac{1}{2}}} = 2.\frac{1}{2} = 1.\)d) \({\log _{ab}}\left( {a\sqrt b } \right) = {\log _{a.{a^{\frac{1}{2}}}}}\left( {a.{a^{\frac{1}{4}}}} \right) = {\log _{{a^{\frac{3}{2}}}}}{a^{\frac{5}{4}}} = \frac{2}{3}.{\log _a}{a^{\frac{5}{4}}} = \frac{2}{3}.\frac{5}{4} = \frac{5}{6}.\)