Given that:
$a^x=b ….(1)$
$b^y=c ….(2)$
$c^z=a….(3)$
taking log in above equations we get:
$\implies xlog_2 a=log_2b$
$\implies$ $x=\frac{log_2 b}{log_2 a}….(4)$
in the same way
$y=\frac{log_2 c}{log_2 b}….(5)$
$z=\frac{log_2 a}{log_2 c}…..(6)$
multiply equations (4),(5),(6).
$\implies x*y*z=\frac{log_2b}{log_2a}*\frac{log_2c}{log_2b}*\frac{log_2a}{log_2c}$
$\implies x*y*z=1$
Option (C) is correct.