If $1,{\log }_y{}^x,{\log }_z{}^y,-15{\log }_x{}^z$  are in $A.P$ , then

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

${\log }_y{}^x=1+d\Rightarrow x=y^{1+d}$

${\log }_z{}^y=1+2d\Rightarrow y=z^{1+2d}$

$-15{\log }_x{}^z=1+3d\Rightarrow z=x^{-\left(1+3d\right)/15}$

$x=y^{1+d}=z^{\left(1+2d\right)\left(1+d\right)}$

$=x^{-\left(1+d\right)\left(1+2d\right)\left(1+3d\right)/15}$

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

$6d^3+11d^2+6d+6=0$

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

$x=y^{1+d}=y^{-1}$

$y=z^{1+2d}=z^{-3}$

$x={\left(z^{-3}\right)}^{-1}=z^3$