$\frac{1}{Log_({p/q}) x} + \frac{1}{Log_({q/r}) x} +\frac{1}{Log_({r/p}) x}$
It is known that :-
${Log_{a} x}$ can be written as $\frac{Log_{x}x}{Log_{x} a}$.
Similarly , the above equation can be written as :-
${Log_{x} \frac{p}{q}}$ + ${Log_{x} \frac{q}{r}}$+${Log_{x} \frac{r}{p}}$ = ${Log_{x} (\frac{p}{q}*\frac{q}{r}*\frac{r}{p}})$ = $Log_{x}1$ = 0