Question

If $x,y,z$ are in H.P. then the value of $\log (x+z)+\log (x-2y+z)$

• A. $2\log |(x-z)|$
• B. $2\log |(y-z)|$
• C. $4\log |(x-z)|$
• D. $log|(y-z)|$

Answer 1

The correct option is A $2\log |(x-z)|$

$\because x,y,zϵ$ A.P $\Rightarrow y=\frac{2xz}{x+z}..\left(1\right)$
Now $\log (x+z)+\log (x-2y+z)=\log (x+z)+\log (x+z-\frac{4xz}{x+z})$ using (1)
$=\log (x+z)+\log \left(\right(x+z{)}^2-4xz)-\log (x+z)=\log (x-z{)}^2=2\log |x-z|$
Hence, option 'A' is correct.