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log2 3和log3 4比大小

1)根据换底公式 log2的3=log6的3/log6的2 log3的4=log6的4/log6的3 2)相减 (log6的9log6的8)/(log6的2*log6的3) 大于 0 所以 前者 大于 后者

log2(3)大于log3(4),换底公式相除,log2(3)等于lg3/lg2,log3(4)等于lg4/lg3,做比较发现,后者比前者小于1

log2(3)=1+log2(3/2)log3(4)=1+log3(4/3)log3(4)如果学过基本不等式log2(3)>0log3(4)>0log2(3)/log3(4)=lg3/lg2/(lg4/lg3)=lg3/(lg2lg4)lg2lg41所以 log2(3)>log3(4)

就用除法吧、减法复杂

g2(3)=1+log2(3/2)log3(4)=1+log3(4/3)<1+log2(4/3)<1+log2(3/2)=log2(3)所以 log2(3)>log3(4)如果学过基本不等式log2(3)>0log3(4)>0log2(3)/log3(4)=lg3/lg2/(lg4/lg3)=lg3/(lg2lg4)lg2lg4<[(lg2+lg4)/2]=[(lg8)/2]<[lg9/2]=(lg3)所以log2(3)/log3(4)>1所以 log2(3)>log3(4)

是log2 3和log3 4吗? log2 3=lg3/lg2 log3 4=lg4/lg3=2lg2/lg3 log2 3-log3 4 =lg3/lg2-2lg2/lg3 =[(lg3)^2-2(lg2)^2]/lg2 *lg3 =(lg3+根号2*lg2)(lg3-根号2*lg2)/lg2 *lg3 =[(lg3+根号2*lg2)lg3/2根号2次方]/lg2 *lg3 因为2根号2次方<3,所以log2 3-log3 4>0 即log2 3>log3 4

换底公式log2(3)=lg3-lg2log3(4)=lg4-lg3log4(5)=lg5-lg4log2(3)-log3(4)=lg3-lg2-(lg4-lg3)=2lg3-lg2-lg4=lg9-lg8>0即log2(3)大于log3(4)同理log3(4)大于log4(5)

2>1log2(x)递增4/3>1log2(4/3)>log2(1)=0同理log0.2(x)递减所以log0.2(4/3)<log0.2(1)=0所以log2(4/3)>log0.2(4/3)>

lgx在(0,+∞)上为增函数,23

log2(3)>log2(2)=log3(3)>log3(2)

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