若2x=3y=5z,且x,y,z都是正数,则2x,3y,5z从小到大依次为______.
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若2x=3y=5z,且x,y,z都是正数,则2x,3y,5z从小到大依次为______.
令2x=3y=5z=t,则t>1,x=
lgt
lg2,y=
lgt
lg3,z=
lgt
lg5,
∴2x−3y=
2lgt
lg2−
3lgt
lg3=
lgt•(lg9−lg8)
lg2•lg3>0,∴2x>3y;
同理可得:2x-5z<0,∴2x<5z,∴3y<2x<5z.
故答案为:3y<2x<5z
lgt
lg2,y=
lgt
lg3,z=
lgt
lg5,
∴2x−3y=
2lgt
lg2−
3lgt
lg3=
lgt•(lg9−lg8)
lg2•lg3>0,∴2x>3y;
同理可得:2x-5z<0,∴2x<5z,∴3y<2x<5z.
故答案为:3y<2x<5z
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