Question

If x=1+logabc, y=1+logbac, z=1+logcab, Prove that xy+yz+zx = xyz

Answer 1

"x = 1 + loga bc can be written as 1 + log bc / log a which again after calculation comes as

(log a + log bc ) / log a = log abc / log a => log a abc

means  x= log abc to base a

same way

y = log abc to base b

z = log abc to base c

1/x = log a to base abc

1/y = log b to base abc

1/z = log c to base abc

here base is now common , just when we add 1/x + 1/y + 1/z = xy + yz + zx / xyz

we will calculate this value

1/x + 1/y + 1/z = log abc a + log abc b + log abc c

(xy + yz + zx ) /xyz = log abc abc

(xy + yz + zx ) /xyz = 1

xy + yz + zx  = xyz

hence proved"