Question

If x,y,z are in H.P. prove that
$\log (x+z)+\log (x+z-2y)=2\log (x-z)$

Answer 1

Since x,y,z are in H.P. $\therefore y=\frac{2xz}{x+z}$
$\therefore x-2y+z=x+z-\frac{4xz}{x+z}$
$=\frac{(x+z{)}^2-4xz}{x+z}=\frac{(x-z{)}^2}{x+z}$
$\therefore$ $\log (x-2y+z)=2\log (x-z)-\log (x+z)$
or $\log (x+z)+\log (x-2y+z)=2\log (x-z)$