Question

if $1,{\log }_{y}x,{\log }_{z}y,-15{\log }_{x}z$ are in AP,then ?

• A. $z^3=x$
• B. $x=y^{-1}$
• C. $z^{-3}=y$
• D. All of these

Answer 1

The correct option is A $z^3=x$

If $1,{\log }_{y}x,{\log }_{z}y,(-15{\log }_{x}z)$ are in $AP$

Let $d$ be the common difference

then, ${\log }_{y}x=1+d\Rightarrow x=y^{1+d}{\log }_{z}y=1+2d\Rightarrow y=z^{1+2d}(-15{\log }_{x}z)=1+3d\Rightarrow z=x^{\frac{-(1+3d)}{15}}\because x=y^{1+d}\Rightarrow x=z^{(1+2d)(1+d)}\Rightarrow x=x^{\frac{-(1+d)(1+2d)(1+3d)}{15}}\Rightarrow \frac{-(1+d)(1+2d)(1+3d)}{15}=1\Rightarrow (d+2)(6d^2-d+8)=0\Rightarrow d=-2$

the value of $d$ in equation $(6d^2-d+8)=0$ is not real because the value of determinant is less than zero.

hence, $x=y^{1-2}=y^{-1}=z^{-(1+2(-2\left)\right)}=z^3$