Question

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

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

Answer 1

The correct option is B $2log(x-z)$

$\begin{array}{c}Accordingtoquestion:\\ log(x+z)+log(x-2y+z)=|\begin{array}{c}given:x,y,z-H.P\\ y=\frac{2\times z}{x+z}\end{array}\\ \Rightarrow log(x+z)+log\left(x+z-2\times \frac{2\times z}{x+z}\right)\\ \Rightarrow log(x+z)+log\left(\frac{{(x+z)}^2-4xz}{x+z}\right)|weknow:\log \left(\frac{a}{b}\right)=\log a-\log b\\ \Rightarrow log(x+z)+log\left[{(x+z)}^2-4xz\right]-\log (x+z)\\ \Rightarrow \log \left[x^2+z^2+2xz-4xz\right]\\ \Rightarrow \log \left[x^2+z^2-2xz\right]\\ \Rightarrow \log (x-z{)}^2|\log a^b=b\log a\\ \therefore 2\log (x-z)\end{array}$

So, that the correct option is B,.