已知x,y,z都是正数,且3^x=4^y=6^z求证1/z-1/x=1/2y

Question

已知x,y,z都是正数,且3^x=4^y=6^z
求证 1/z-1/x=1/2y

Answer 1

∵x,y,z都是正数
3^x=4^y=6^z
∴lg(3^x)=lg(4^y)=lg(6^z)
即：xlg3=ylg4=zlg6
设xlg3=ylg4=zlg6=k
则x=k/lg3
y=k/lg4
z=k/lg6
∴1/z-1/x
=1/(k/lg6)-1/(k/lg3)
=lg6/k-lg3/k
=(lg6-lg3)/k
=lg2/k
∴1/2y
=1/(2k/lg4)
=lg4/(2k)
=2lg2/(2k)
=lg2/k
∴1/z-1/x=1/2y