Question

已知x,y,z大于零,3的x次方=4的y次方=6的z次方,求1/z-1/x=1/2y;比较3x;4y;6z的大小

Answer 1

设3^x=4^y=6^z=k.(k>1)

x=log3(k)

y=log4(k)

z=log6(k)

1/z-1/x=logk(6)-logk(3)=logk(2)

1/(2y)=1/2*logk(4)=logk(2)

故:1/z-1/x=1/(2y)

1/(3x)=1/3*logk(3)=logk(3^1/3)

1/(4y)=1/4*logk(4)=logk(4^1/4)

1/(6z)=1/6*logk(6)=logk(6^1/6)

由于logk(3^1/3)>logk(4^1/4)>logk(6^1/6)

所以,1/3x>1/4y>1/6z

即3x