If p^a=q^b=r^c and pqr=1, prove that 1/a+1/b+1/c=0

Let p^a=q^b=r^c=k
then,log k=logp^a
so,log k=alog p
log k/log p=a-------------------------1
       log k=log q^b
so log k=blog q
log k/log q=b---------------------2
       log k=log r^c
so log k=clog r
log k/log r=c------------------------3

thus 1/a+1/b+1/c=log p/log k + log q/log k +log r/log k  (from 1,2,3)
                          =(log p +log q+log r )/log k
                         =log(pqr)/log k                       
                         =0/log k              (as given  pqr is 1)
                         =0  ,hence proven