Question

If $x,y,z$ are in $H.P$, then $\log |x+z|+\log |x-2y+z|$ is equal to

• A. $\log |x-z|$
• B. $2\log |x-z|$
• C. $2\log |x+z|$
• D. none of these

Answer 1

The correct option is B $2\log |x-z|$

Given, $x,y,z$ are in H.P.
$\Rightarrow y=\frac{2xz}{x+z}$
$\Rightarrow (x+z)y=2xz...\left(1\right)$

$\log |x+z|+\log |x-2y+z|$
$=\log |(x+z)(x+z-2y)|$
$=\log |(x+z{)}^2-2y(x+z)|$
$=\log |(x+z{)}^2-4xz|[\text{From}\left(1\right)]$
$=\log (x-z{)}^2=2\log |x-z|$