NIELIT 2019 Feb Scientist D - Section C: 26

Question

If $a^{x}=b$, $b^{y}=c$ and $c^{z}=a$, then the value of $xyz$ is :

• A. $0$
• B. $1$
• C. $\frac{1}{3}$
• D. $\frac{1}{2}$

Answer 1

$\textrm{Given;}$

$a^{x}=b$,

$b^{y}=c$,

$c^{z}=a$

$\textrm{Taking log in both side we get:}$

$\Leftrightarrow$ $a^{x}=b$

$\Leftrightarrow$ $\log{a^{x}}$=$\log{b}$

$\Leftrightarrow$ $x.{\log{a}}$=$\log{b}$

$\Leftrightarrow$ $\textrm{$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.