Không sử dụng máy tính cầm tay, hãy tính:
a) \(\log {}_3\sqrt[4]{3};\)
b) \({4^{{{\log }_2}3}}\);
c) \({27^{{{\log }_9}2}}\);
d) \({9^{{{\log }_{\sqrt 3 }}2}}\).
Áp dụng:
a) \(\sqrt[n]{{{a^m}}} = {a^{\frac{m}{n}}}\); \({\log _a}{a^b} = b\).
b, c, d) \({a^{{{\log }_a}b}} = b\) và \({\left( {{a^b}} \right)^c} = {\left( {{a^c}} \right)^b}\)
a) \(\log {}_3\sqrt[4]{3} = {\log _3}\left( {{3^{\frac{1}{4}}}} \right) = \frac{1}{4}\)
b) \({4^{{{\log }_2}3}} = {\left( {{2^2}} \right)^{{{\log }_2}3}} = {\left( {{2^{{{\log }_2}3}}} \right)^2} = {3^2} = 9\)
c) \({27^{{{\log }_9}2}} = {\left( {{3^3}} \right)^{{{\log }_9}2}} = {\left( {{3^{{{\log }_9}2}}} \right)^3} = {\left( {{3^{\frac{1}{2}{{\log }_3}2}}} \right)^3} = {\left( {{3^{{{\log }_3}2}}} \right)^{\frac{3}{2}}} = {2^{\frac{2}{3}}}\)
d) \({9^{{{\log }_{\sqrt 3 }}2}} = {\left( {{{\sqrt 3 }^4}} \right)^{{{\log }_{\sqrt 3 }}2}} = {\left( {{{\sqrt 3 }^{{{\log }_{\sqrt 3 }}2}}} \right)^4} = {2^4} = 16\)