Question

if x=logab base c y=logbc base a and z=logca base b prove that 1/1+X+1/1+y+1/1+z=1

Answer 1

1+x = logc base c +logab base c=logabc base c
1+y = loga base a +logbc base a=logabc base a
1+z = logb base b+logca base b=logabc base b
NOW,

1/1+x = lnc/lnabc {base change formula}....(1)
1/1+y = lna/lnabc ...........(2)
1/1+z = lnb/lnabc. ............(3)
sum of 1,2&3
lnabc/lnabc= 1 PROVED