IF (2x)^ln 2=(3y)^ln 3, and 3^ln x=2^ln y, then x equals :

Answers (1)

Solution:        (2x)^ln 2=(3y)^ln 3      .......(1)

                      3^ln x=2^ln y              .......(2)

                      From (2)  , we have

                       x^ln 3=y^ln 2            ........(3)

                    From   (1)  ,     (ln2)[ln2+ln x]=(ln 3)[ln 3+ln y],

                   (ln x)^2-(ln 3)^2+(ln 2)(ln x)=(ln 3)(ln y)     ........(4)

                   From   (3)   , we have

                                      (ln 3)(ln x)=(ln 2)(ln y).     .......(5)

                     Dividing (4) by (5) , we get

                 (ln x)[(ln 3)^2-(ln 2)^2]=-ln 2[(ln 3)^2-(ln 2)^2],

           Rightarrow                          (ln x)=ln frac12Rightarrow x=frac12

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