Question

If a^x=c^q=b and c^y=a^z=d, then
1) xy=qz
2) x/y=q/z
3) x+y=q+z
4) x-y=q-z
Please explain..

Answer 1

If a^x=c^q=b and c^y=a^z=d, 
a^x = b
taking log on both side
x log a = log b
x = log b / log a
similarly, y = log d / log c
              q = log b / log c &  z = log d / log a
1) x y = ( log b x log d ) / (log a x log c) ---> LHS

   q z  = (log b x log d) / ( log a x log c) ---->  RHS

       so, xy = qz

2) similarly x / y = q /z

3)  x + y =  log b / log a + log d / log c
              = (log b x log c + log d x log a) / log a x log c (taking LCM)
     q+z = log b / log a + log d / log c
              = (log b x log c + log d x log a) / log a x log c (taking LCM)

  so x+y = q+z
4) similarly x - y = q - z