Logarithmic Equations

 

 

A logarithmic equation is an equation that contains a logarithm of variable quantity. The following property is useful:

 

       For two logarithms of the same base:

                                       

 

The laws of logarithms are important in this section.

 

 

Solve 2 log₃x - log₃ (2x - 3) = 1 + log₃9

 

   Solution:

 

              2 log₃x - log₃ (2x - 3) = 1 + log₃9

           log₃x² - log₃ (2x - 3) = log₃3 + log₃9

                                   

                                            

                    x² - 54x + 81 = 0

                                                     

                                                     

                                       = 52.5 or 1.54    approximated to 3 significant figures

 

 

Solve the following equation:

                            log_x27 - log₃x = -2

 

Solution:

 

                            log_x27 - log₃x = -2

        log₃27 / log₃x - log₃x = -2

        (log₃x)² - 2 log₃x + 3 = 0

       (log₃x - 3) (log₃x + 1) = 0

                        log₃x = 3        or     log₃x = -1

                             x = 27                    x = 1/3

 

 

Given

              ,

                           show that a² + b² = 23ab.

 

Solution:

                 

                                   

                                       

                        25ab = a² + 2ab + b²

                         a² + b² = 23ab               shown

 

 

Given that log_kx²y³ = 5 and log_k(x/y) = 2, find the numerical value of

          .

 

Solution:

 

                      log_kx²y³ = 5

          2 log_kx + 3 log_ky = 5      ------(1)

 

                     log_k(x/y) = 2

                log_kx - log_ky = 2      ------(2)

 

   (1) + 3 (2) :  5 log_kx  = 11

                          log_kx = 11/5

 

   (1) - 2 (2) :   5 log_ky = 1

                         log_ky = 1/5

 

         

                    = 1/4 log_kxy

                    = 1/4 (log_kx +log_ky)

                    = 1/4 (11/5 + 1/5)

                    = 3/5