If a=b^x, b=c^y and c = a^z prove that xyz

=1

If ax=b and by=c, substituting you get that axy=c. Now, if cz=a, substituting again, axyz=a, from where it follows that xyz=1.

or

c = {b^y} = [ (a^x) ^y] = {a^ (xy) }

a = [c^z] = [{a^ (xy) }^z] = [a^ (xyz) ]

(a^1) = [a^ (xyz) ]

Therefore, (xyz) = 1.