Question

Вычислите:
1. $frac{log _{4} 36}{log _{4}6 }$
2. $frac{log _{0,3}32 }{log _{0,3} 64}$
3. $frac{2log _{0,5}2+log _{0,5} sqrt{10} }{ log_{0,5} 10- log_{0,5} sqrt{10} + log_{0,5} 4}$
4. $frac{ log_{0,3}16 }{ log_{0,3} 15- log_{0,3} 30}$

Answer 1

$frac{log _{4}36}{log _{4}6} = frac{log _{4}6^2}{log _{4}6} = frac{2log _{4}6}{log _{4}6} = 2;$

$frac{log _{0,3}32}{log _{0,3}64} = frac{log _{0,3}2^5}{log _{0,3}2^6} = frac{5log _{0,3}2}{6log _{0,3}2} = frac{5}{6};$

$frac{2log _{0,5}2+log _{0,5}sqrt{10}}{log_{0,5}10-log_{0,5}sqrt{10}+log_{0,5}4} = frac{log _{0,5}(2^2cdotsqrt{10})}{log_{0,5} frac{10cdot4}{sqrt{10}} } = frac{log _{0,5}(4cdotsqrt{10})}{log _{0,5}(4cdotsqrt{10})} = 1;$

$frac{log_{0,3}16}{log_{0,3}15-log_{0,3}30} = frac{log_{0,3}2^4}{log_{0,3}frac{15}{30}} = frac{4log_{0,3}2}{log_{0,3}2^{-1}} = frac{4log_{0,3}2}{-log_{0,3}2}} = -4.$