In the one-variable case we can take a log transform of the function f ( x ) → log ⁡ ( f ( x ) ) and know that the same value maximizes both because log is increasing. The log turns products into sums which facilitates taking derivatives. Can the same argument be applied for multivariable cases? The question is: for a function f ( x , y ) if I wish to find the x ⋆ , y ⋆ that maximize f ( x , y ) are these the same x ⋆ ⋆ , y ⋆ ⋆ that maximize log ⁡ ( f ( x , y ) )? My intuition is yes, nothing should be different in this multivariable case, but I desire a quick sanity check before setting off with computations. I would appreciate a rigorous proof or a counterexample.

Question

In the one-variable case we can take a   log transform of the function   f  (  x  )  →  log  ⁡  (  f  (  x  )  ) and know that the same value maximizes both because log is increasing. The log turns products into sums which facilitates taking derivatives. Can the same argument be applied for multivariable cases? The question is: for a function   f  (  x  ,  y  ) if I wish to find the       x    ⋆    ,      y    ⋆   that maximize   f  (  x  ,  y  ) are these the same       x          ⋆      ⋆        ,      y          ⋆      ⋆       that maximize   log  ⁡  (  f  (  x  ,  y  )  )? My intuition is yes, nothing should be different in this multivariable case, but I desire a quick sanity check before setting off with computations. I would appreciate a rigorous proof or a counterexample.

heilaritikermx · Accepted Answer

Yes (as long as   f takes positive values) since   f  (      x    ∗    ,      y    ∗    )  ≥  f  (  x  ,  y  ) if and only if   log  ⁡  f  (      x    ∗    ,      y    ∗    )  ≥  log  ⁡  f  (  x  ,  y  ).

In the one-variable case we can take a log transform of the function f ( x ) →

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