Laws of Logarithms

 

 

Now that you have understood the definition of logarithms, let us take a look at the manipulation of logarithms:

 

        Product Law of Logarithms:

                          log_ab + log_ac = log_abc

            Quotient Law of Logarithms:

                            log_ab - log_ac = log_a(b/c)

              Power Law of Logarithms:

                         log_ab^x = x log_ab

 

As in the laws of surds, the laws of logarithms are derived from the laws of indices. In the above identities, a, b and c must be positive.

 

        Special logarithms:

                         log_aa = 1

                         log_a1 = 0

 

           Change of base formula:

                                   

 

 

    Proof of change of base formula:

            Let x = log_ab,    a^x = b

                       log_ca^x = log_cb

                     x log_ca = log_cb              power law of logarithms

                            

                     

          Special case:

                      

   Another useful property of logarithms is:

                      

 

   Proof of property:

                   Let  = x

                         log_ab = log_ax            convert to logarithmic form

                               x = b

                        

 

 

The next page deals with examples on the application of the laws.