Question

已知正实数xyz满足3的x次方=4的y次方=6的z次方,求证:1/z-1/x=1/2y

Answer 1

3^x=4^y=6^z

ln(3^x)=ln(4^y)=ln(6^z)

xln3=yln4=zln6

xln3=2yln2=z(ln2+ln3)

设xln3=2yln2=z(ln2+ln3)=t

ln3=t/x,ln2=t/(2y),ln2+ln3=t/z

t/(2y)+t/x=t/z

1/(2y)+1/x=1/z

1/z-1/x=1/(2y)