Question

If $1,{\log }_{y}x,{\log }_{z}y,-15{\log }_{x}z$ are in $A.P.$ , then

• A. $z^3=x$
• B. $x=y^{-1}$
• C. $z^{-3}=y$
• D. $x=y^{-1}=z^3$
• E. All the above

Answer 1

The correct option is E All the above

Let $d$ be the common difference of the $A.P.$

Then,

${\log }_{y}x=1+d$

$\Rightarrow$ $x=y^{1+d}$ ----- ( 1 )

${\log }_{z}y=1+2d$

$\Rightarrow$ $y=z^{1+2d}$ ------ ( 2 )

$-15{\log }_{x}z=1+3d$

$\Rightarrow$ $z=x^{\frac{-(1+3d)}{15}}$ ------ ( 3 )

$x=y^{1+d}=z^{(1+2d)(1+d)}=x^{\frac{-(1+d)(1+2d)(1+3d)}{15}}$

$\Rightarrow$ $(1+d)(1+2d)(1+3d)=-15$

$\Rightarrow$ $6d^3+11d^2+6d+16=0$

$\Rightarrow$ $(d+2)(6d^2-d+8)=0$

$\Rightarrow$ $d=-2$

Substituting value of $d$ we get,

$\Rightarrow$ $x=y^{-1}=z^3=x^{\frac{1}{3}}$ or

$\Rightarrow$ $x=y^{-1}=z^3,y=z^{-3}$