Laws of Logarithms

1. log_bMN = log_bM + log_bN
   Proof:
   Let M = b^x and N = b^y
   Then MN = (b^x)(b^y)
           MN = b^{x + y}
   Write as a logarithm:
   log_bMN = x + y
   Since M = b^x, then log_bM = x
   Since N = b^y, then log_bN = y
   And log_bMN = log_bM + log_bN
   
   
2. log_b(M/N) = log_bM - log_bN
   Proof:
   Let M = b^x and N = b^y
   Then M = b^x
           N     b^y
           M = b^{x - y}
             N
   Write as a logarithm:
   log_b(M/N) = x - y
   Since M = b^x, then log_bM = x
   Since N = b^y, then log_bN = y
   And log_b(M/N) = log_bM - log_bN
   
   
3. log_bM^k = k log_bM
   Proof:
   Let M = b^x
   Then M^k = (b^x)^k
           M^k = b^xk or b^kx
   Write as a logarithm:
   log_bM^k = kx
   Since M = b^x, then log_bM = x
   And log_bM^k = k log_bM
   
   
4. log_bM = log_bN if and only if M = N
   
   
5. log_xM = log_yM if and only if x = y

Problems