$\textrm{Given;}$
$a^{x}=b$,
$b^{y}=c$,
$c^{z}=a$
$\textrm{Taking log in both side we get:}$
$\Leftrightarrow$ $a^{x}=b$
$\Leftrightarrow$ $\log{a^{x}}$=$\log{b}$
$\Leftrightarrow$ $x.{\log{a}}$=$\log{b}$
$\Leftrightarrow$ $\textrm{$x$ = $\frac{\log{b}}{\log{a}}$}$
$\textrm{In same way}$
$\Leftrightarrow$ $\textrm{$y$ = $\frac{\log{c}}{\log{b}}$}$
$\Leftrightarrow$ $\textrm{$z$ = $\frac{\log{a}}{\log{c}}$}$
$x*y*z$=$\textrm{$\textrm{( $\frac{\log{b}}{\log{a}}$}$)*($\textrm{ $\frac{\log{c}}{\log{b}}$}$ )*($\textrm{$\frac{\log{a}}{\log{c}}$}$ ) }$
$\Leftrightarrow$ $x*y*z$$=$ $1$
Option $B$ is correct.