2017/014) if $x,y,z$ are in HP then show that $\log(x+z) + \log(x+z-2y) + 2 \log(x-z)$(IIT-1978)

let us eliminate y from LHS as RHS does not contain y
We have $x,y,z$ are in HP
hence $\frac{1}{x} + \frac{1}{z} = \frac{2}{y}$
or $y(z+x) = 2xz$
or $y=\frac{2xz}{x+z}$
hence $x+z-2y = x+ z - \frac{4xz}{x+z}$
or $(x+z-2y)(x+z) = (x+z)^2 - 4xz = x^2+z^2+ 2xz - 4xz$
$= x^2+z^2 - 2xz = (x-z)^2$
taking log on both sides we get $\log(x+z) + \log(x+z-2y) + 2 \log(x-z)$