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比较:2^(1/2) 和 (2/3)^(-1) 和3^(1/3)的 大小
令,m1=2^(1/2),m2=(2/3)^(-1),m3=3^(1/3)
log2(m1)=1/2,
log2/3(m2)=-1,
log2(m2)=(-1)[log2(2)-log2(3)=log2(3)-1,
log3(m3)=1/3,
log2(m3)=1/3log2(3).
log2(m2)-log2(m3)=log2(3)-1-1/3log2(3)
=2/3log2(3)-1=2/3[log2(3)-3/2log2(2)]
=2/3[log2(3/2√2)]
∵3/2√2>1,∴log2(3/2√2)>0,
log2(m2)-log2(m3)>0,
log2(m2)>log2(m3),
log2(m3)-log2(m1)=1/3log2(3)-1/2=1/3[log2(3)-3/2log2(2)]
=1/3[log2(3)-log2(2)^3/2]
=1/3[log2(3/2√2)]
∵3/2√2>1,∴log2(3/2√2)>0,
log2(m3)-log2(m1)>0,
log2(m3)>log2(m1).
∴(2/3)^(-1)>3^(1/3)>2^(1/2) .
1年前
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