Separating the log of a sumI know there is no formula to separate the log of a sum, e.g. log ⁡ ( X + Y ) into two parts, but are there any approximation rules that can allow me to achieve this objective? E t ( 1 + r t + 1 K ) = E t [ 1 X t + 1 α A 0 Y t + 1 K t + 1 + Q t + 1 ( 1 − δ ) Q t ] Suppose we ignore the expectations operator for the moment.

Question

Separating the log of a sumI know there is no formula to separate the log of a sum, e.g.   log  ⁡  (  X  +  Y  ) into two parts, but are there any approximation rules that can allow me to achieve this objective?      E    t    (  1  +      r          t      +      1        K    )  =      E    t        [                                        1                          X                              t                +                1                                              α                      A            0                                              Y                              t                +                1                                                    K                              t                +                1                                              +                      Q                          t              +              1                                (          1          −          δ          )                          Q          t                      ]  Suppose we ignore the expectations operator for the moment.

megagoalai · Accepted Answer

log  ⁡  (  X  +  Y  )  =  log  ⁡  (  X  )  +  log  ⁡  (  1  +  Y      /    X  ), so if either   X is small compared to   Y or vice versa then you can approximate with a Taylor approximation.