$\color{maroon}{pqr \neq 0}$
$\color{maroon}{p^{-x} = \dfrac{1}{q}}$
Or,$\log(p^{-x}) = \log(\dfrac{1}{q})$
Or, $-x \log (p) = \log(1)-\log(q)$ $\qquad\Big[∵\color{blue}{\log_b(m^n)= n. \log_b(m)\\\qquad\qquad \log_b(\dfrac{m}{n})= \log_b(m)-\log_b(n)} \Big]$
Or, $x\log(p) = \log (q)$ $\qquad[∵\color{blue}{\log(1)=0}]$
Or, $\color{green}{x= \dfrac{\log(q)}{\log(p)}}$
$\color{maroon}{q^{-y} = \dfrac{1}{r}}$
Or,$\log(q^{-y}) = \log(\dfrac{1}{r})$
Or, $-y \log (q) = \log(1)-\log(r)$ $\qquad\Big[∵\color{blue}{\log_b(m^n)= n. \log_b(m)\\\qquad\qquad \log_b(\dfrac{m}{n})= \log_b(m)-\log_b(n)}\Big]$
Or, $y\log(q) = \log (r)$ $\qquad[∵\color{blue}{\log(1)=0}]$
Or, $\color{green}{y= \dfrac{\log(r)}{\log(q)}}$
$\color{maroon}{r^{-z} = \dfrac{1}{p}}$
Or,$\log(r^{-z}) = \log(\dfrac{1}{p})$
Or, $-z \log (r) = \log(1)-\log(p)$ $\qquad\Big[∵\color{blue}{\log_b(m^n)= n. \log_b(m)\\\qquad\qquad \log_b(\dfrac{m}{n})= \log_b(m)-\log_b(n)}\Big]$
Or, $z\log(r) = \log (p)$ $\qquad[∵\color{blue}{\log(1)=0}]$
Or, $\color{green}{z= \dfrac{\log(p)}{\log(r)}}$
∴ $\color{black}{x \times y \times z}$ = $ \dfrac{\log(q)}{\log(p)} \times \dfrac{\log(r)}{\log(q)} \times \dfrac{\log(p)}{\log(r)}$
$\qquad \qquad = \color{black}{1}$
Correct Answer: $C$