Question

If $x,y,z$ are in HP, then the value of the expression $\log (x+z)+\log (x-2y+z)$ will be:

• A. $\log (x-z)$
• B. $2\log (x-z)$
• C. $3\log (x-z)$
• D. $4\log (x-z)$

Answer 1

The correct option is B

$2\log (x-z)$

Explanation for the correct option.

If $x,y,z$ are in HP, then

$\begin{array}{rcl}\frac{2}{y}& =& \frac{1}{x}+\frac{1}{z}\\ & \Rightarrow & y=\frac{2xz}{z+x}\end{array}$

Then, $\log (x+z)+\log (x-2y+z)$

$$\begin{gathered} =\log \left(x+z\right)+\log \left[x+z-2\left(\frac{2xz}{x+z}\right)\right] \\ =\log \left(x+z\right)+\log \left[\frac{{\left(x+z\right)}^2-4xz}{x+z}\right] \\ =\log \left(x+z\right)+\log \left[{\left(x+z\right)}^2-4xz\right]-\log \left(x+z\right)\left[by\log \left(\frac{a}{b}\right)=\log a-\log b\right] \\ =\log {\left(x-z\right)}^2 \\ =2\log \left(x-z\right)\left[by\log a^2=2\log a\right] \end{gathered}$$

Hence, option B is correct.