题目内容
12.比较下列各数的大小:20.7,log54,log${\;}_{\frac{1}{3}}$5,log3$\frac{1}{4}$,log4$\frac{1}{3}$.分析 先判断20.7>20=1,0<log54<1,再判断${log}_{\frac{1}{3}}$5<log3$\frac{1}{4}$<-1,log4$\frac{1}{3}$>-1,从而得出正确的结论.
解答 解:∵20.7>20,
∴20.7>1,
又∵log51<log54<log55,
∴0<log54<1,
∴20.7>log54>0;
又∵${log}_{\frac{1}{3}}$5=-${log}_{\frac{1}{3}}$$\frac{1}{5}$,
log3$\frac{1}{4}$=-${log}_{\frac{1}{3}}$$\frac{1}{4}$,
且$\frac{1}{5}$<$\frac{1}{4}$<$\frac{1}{3}$,
∴${log}_{\frac{1}{3}}$$\frac{1}{5}$>${log}_{\frac{1}{3}}$$\frac{1}{4}$>${log}_{\frac{1}{3}}$$\frac{1}{3}$,
∴-${log}_{\frac{1}{3}}$$\frac{1}{5}$<-${log}_{\frac{1}{3}}$$\frac{1}{4}$<-${log}_{\frac{1}{3}}$$\frac{1}{3}$=-1,
又∵0=log41>log4$\frac{1}{3}$>log4$\frac{1}{4}$=-1,
∴0>log4$\frac{1}{3}$>log3$\frac{1}{4}$>${log}_{\frac{1}{3}}$5;
综上,20.7>log54>log4$\frac{1}{3}$>log3$\frac{1}{4}$>${log}_{\frac{1}{3}}$5.
点评 本题考查了利用函数的单调性判断数值大小的应用问题,是基础题目.
A. | $\frac{1}{2}$ | B. | -$\frac{1}{2}$ | C. | $\frac{\sqrt{3}}{2}$ | D. | -$\frac{\sqrt{3}}{2}$ |
A. | [0,$\frac{4}{3}$] | B. | [$\frac{1}{2}$,2) | C. | [$\frac{1}{2}$,$\frac{4}{3}$] | D. | [$\frac{1}{2}$,+∞) |