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$\textrm{Given that: }$

$a^x=b ….(1)$

$b^y=c ….(2)$

$c^z=a….(3)$

$\textrm{taking log in above equations we get:}$

$\implies$ $xlog_2a=log_2b$

$\implies$ $x=\frac{log_2b}{log_2a}….(4)$

$\textrm{in the same way}$

$y=\frac{log_2c}{log_2b}….(5)$

$z=\frac{log_2a}{log_2c}…..(6)$

$\textrm{multiply equations (4),(5),(6).}$

$\implies$ $x*y*z=\frac{log_2b}{log_2a}*\frac{log_2c}{log_2b}*\frac{log_2a}{log_2c}$

$\implies$ $x*y*z=1$

$\textrm{Option C is correct.}$

$a^x=b ….(1)$

$b^y=c ….(2)$

$c^z=a….(3)$

$\textrm{taking log in above equations we get:}$

$\implies$ $xlog_2a=log_2b$

$\implies$ $x=\frac{log_2b}{log_2a}….(4)$

$\textrm{in the same way}$

$y=\frac{log_2c}{log_2b}….(5)$

$z=\frac{log_2a}{log_2c}…..(6)$

$\textrm{multiply equations (4),(5),(6).}$

$\implies$ $x*y*z=\frac{log_2b}{log_2a}*\frac{log_2c}{log_2b}*\frac{log_2a}{log_2c}$

$\implies$ $x*y*z=1$

$\textrm{Option C is correct.}$