Question

Вычислите:
1. $(3lg2+lg0,25):(lg14-lg7)$
2. $( log_{2}12- log_{2}3+ 3^{log _{3}8 } ) ^{lg5}$
3. $( log_{6}2+ log_{6}3+ 2^{ log_{2}4 }) ^{ log_{5}7 }$
4. $frac{ log_{3}63 }{2+ log_{3}7 }$

Answer 1

$(3lg2+lg0,25):(lg14-lg7) = (lg2^3+lg0,25):lgfrac{14}{7} =\= lg(8cdot0,25):lg2 = lg2:lg2 = 1;$

$(log_{2}12- log_{2}3+ 3^{log _{3}8 } )^{lg5} = (log_{2}frac{12}{3} + 8)^{lg5} =\= (log_{2}4+ 8)^{lg5} = (2log_{2}2+ 8)^{lg5} = (2+ 8)^{lg5} = 10^{lg5} = 5;$

$( log_{6}2+ log_{6}3+ 2^{ log_{2}4 }) ^{ log_{5}7 } =( log_{6}(2cdot3)+4) ^{log_{5}7} =(1+4) ^{log_{5}7} =\= 5^{log_{5}7} = 7;$

$frac{log_{3}63 }{2+log_{3}7} = frac{log_{3}63 }{2log_{3}3+log_{3}7} = frac{log_{3}63 }{log_{3}3^2+log_{3}7} = frac{log_{3}63 }{log_{3}(9cdot7)} = frac{log_{3}63 }{log_{3}63} = 1.$