Question

if x, y, z are three positive real numbers then prove that, (y/z)^logyz × (z/x)^logzx ×(x/y)^logxy =1.

Answer 1

LHS = (y/z)^logyz × (z/x)^logzx × (x/y)^logxy

= {y^logyz/z^logyz}{z^logzx/x^logzx}{x^logxy/y^logxy}

y^(logyz) =y^(logy + logz) = y^logy .y^logz
= 1. y^logz

in the same way all
z^logzx = z^logx
z^logyz =z^logy
x^logxy =x^logy
y^logxy =y^logx

= { y^logz /z^logy }{ z^logx /x^logz }{ x^logy/y^logx }


we know,

a^logb = b^loga

so,
y^logz = z^logy
z^logx =x^logz
x^logy =y^logx

so,
= (1 ) × (1) × (1) = RHS