Bài 2. Tính:
a) \({4^{lo{g_2}3}}\);
b) \({27^{lo{g_9}2}}\);
c) \({9^{lo{g_{\sqrt 3 }}2}}\)
d) \({4^{lo{g_8}27}}\);.
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a) \({4^{lo{g_2}3}} = {\left( {{2^2}} \right)^{lo{g_2}3}} = {\left( {{2^{lo{g_2}3}}} \right)^2} = {3^2} = 9\).
b)
\(\eqalign{
& {27^{lo{g_9}2}} = {\left( {{3^3}} \right)^{lo{g_9}2}} = {\left( {{9^{{1 \over 2}}}} \right)^{3lo{g_9}2}} \cr
& = {\left( {{9^{lo{g_9}2}}} \right)^{{3 \over 2}}} = {2^{{3 \over 2}}} = 2\sqrt 2 \cr} \)
c) \({9^{lo{g_{\sqrt 3 }}2}} = {\left( {{{\left( {\sqrt 3 } \right)}^4}} \right)^{lo{g_{\sqrt 3 }}2}} = {\left( {{{\left( {\sqrt 3 } \right)}^{lo{g_{\sqrt 3 }}2}}} \right)^4} = {2^4} \)\(= 16\)
d) Có \({\rm{lo}}{{\rm{g}}_8}{\rm{27 = }}lo{g_{{2^3}}}{3^3} = {3 \over 3}lo{g_2}3 = {\rm{lo}}{{\rm{g}}_2}{\rm{3}}\)
nên \({4^{lo{g_8}27}} = {\left( {{2^2}} \right)^{lo{g_2}3}} = {\left( {{2^{lo{g_2}3}}} \right)^2} = {3^2} = 9\).