设x,y,z∈R+,且3x=4y=6z. (1)求证:1/z−1/x=1/2y; (2)比较3x,4y,6z的大小.

问题描述:

设x,y,z∈R+,且3x=4y=6z
(1)求证:

1
z
1
x
1
2y
;  
(2)比较3x,4y,6z的大小.

(1)证明:设3x=4y=6z=t.∵x>0,y>0,z>0,∴t>1,lgt>0,
x=log3t=

lgt
lg3
y=log4t=
lgt
lg4
z=log6t=
lgt
lg6

1
z
1
x
lg6
lgt
lg3
lgt
lg2
lgt
lg4
2lgt
1
2y

(2)∵3x>0,4y>0,且
3x
4y
3
lgt
lg3
4
lgt
lg4
=log3
4 64
<1

∴3x<4y,同理4y<6z,
故3x<4y<6z.