If `(x(y+z-x))/(logx)=(y(z+x-y))/(logy) =(z(x+y-z))/(logz),p rov et h a tx^y y^x=z^x y^z=x^z z^x`

Explanation

Let `(x(y+z-x))/(log_(a) x) = (y(z+x-y))/(log_(a)y) = (z(x+y-z))/(log_(a) z) = k`
` rArr log_(a) x = (x(y+z-x))/k`
` rArr x = a^((x(y+z-x))/k)`
Similarly,` y = a ^((y(x+z-y))/k) and z=a^((z(x+y-z))/k)`
Now `x^(y)y^(x) = a^ ((xy(y+z-x))/k)a^((yx(z+x-y))/k)`
` = a ^((xy^(2)+xyz-x^(2)y+xyz+x^(2)y-xy^(2))/k)=a^((2xyz)/k)`
Similarly,` z^(y)y^(z) = x^(z)z^(x) = a^((2xyz)/k)`.