Question

If $a^x=b^y=c^z$ and $b^2=ac$ then $\frac{1}{x}+\frac{1}{z}$

• A. $\frac{2}{y}$
• B. $\frac{1}{y}$
• C. $\frac{1}{2}y$
• D. $2y$

Answer 1

The correct option is A $\frac{2}{y}$

$a^x=b^y$

taking log both sides we have

$xloga=ylogb\therefore x=\frac{ylogb}{loga}$

$c^z=b^y$

taking log both sides we have

$zlogc=ylogb\therefore z=\frac{ylogb}{logc}$

Now

$\frac{1}{x}+\frac{1}{y}=\frac{loga}{ylogb}+\frac{logc}{ylogb}=\frac{log\left(ac\right)}{ylogb}$

Now it is given that $b^2=ac$

$\frac{log\left(ac\right)}{ylogb}=\frac{log\left(b^2\right)}{ylogb}=\frac{2}{y}$