solve for x : log (x-5) +log (x+3) = 1

Question

solve for x in this equation : log (x-5) +log (x+3) = 1

All log are to the base of 9

Answer 1

Hi Suama,                      

 

                                           log₉ (x-5) +log₉ (x+3) = 1

 

Since the logs are both base 9 they can be simplified using log properties.

 

                                                               log₉ (x-5) +log₉ (x+3) = log₉9

                                                                     log₉ (x-5)(x+3)=log₉9

                                                                          (x-5)(x+3)=9

 

                                                                         x²-5x+3x-15=9

                                                                         x²-2x-15=9

                                                                         x²-2x-24=0

                                                                        (x-6)(x+4)=0

                                                                      x-6=0           x+4=0

                                                                       x=6              x=-4  Rejected; not in the domain

 

                                                                        Therefore, x=6

 

Check:  log₉ (x-5) +log₉ (x+3) = 1

           log₉ (6-5) +log₉ (6+3) = 1

            log₉1 + log₉9=1

               0 + 1 = 1

                     1 = 1  checks