糖水不等式

$\mathrm{比}\mathrm{较}\mathrm{大}\mathrm{小}{\log }_{6}3,{\log }_{12}6,{\log }_{24}12$

$\mathrm{换}\mathrm{底}\mathrm{公}\mathrm{式}\mathrm{转}\mathrm{变}\mathrm{形}\mathrm{式}$
${\log }_{6}3=\frac{\lg 3}{\lg 6}<1$
${\log }_{12}6=\frac{\lg 6}{\lg 12}=\frac{\lg 3+\lg 2}{\lg 6+\lg 2}$
${\log }_{24}12=\frac{\lg 12}{\lg 24}=\frac{\lg 6+lg2}{\lg 12+\lg 2}$

$0<b<a,m>0,\frac{b}{a}<\frac{b+m}{a+m}$

${\log }_{6}3<{\log }_{12}6<{\log }_{24}12$

$a={\log }_{6}3,b={\log }_{12}6,c={\log }_{24}12$
$\frac{1}{a}={\log }_{3}6=1+{\log }_{3}2$
$\frac{1}{b}={\log }_{6}12=1+{\log }_{6}2$
$\frac{1}{c}={\log }_{12}24=1+{\log }_{12}2$
$0<\frac{1}{c}<\frac{1}{b}<\frac{1}{a}$

$\mathrm{令}f(x)={\log }_{2x}x$
$\mathrm{即}\mathrm{比}\mathrm{较}f(3),f(6),f(12)$
$f(x)\mathrm{为}\mathrm{增}\mathrm{函}\mathrm{数}$