When a>0 and a≠ 1, m >；; 0, N>0, then: (1) log (a) (Mn) = log (a) (m)+log (a) (n); (2)log(a)(M/N)= log(a)(M)-log(a)(N)； (3)log(a)(m n)= nlog(a)(m)(n∈r)(4)log(a n)(m)= 1/nlog(a)(m)(n∈r)(。 0 and b ≠1) (6) a (log (b) n) = n (log (b) a) proof: let a = n x then a (log (b) n) = (n x) log (b) n = log (a) ab = Kloc-0//n)log(aLog (based on a under the root number n) (based on m under the root number n) =log(a)M, log (based on a under the root number n) (based on m under the root number m) = (.