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I tried to do something below

Log[PDF[NormalDistribution[m, s], x]] /. Log[u_[x__] ] :> Plus @@ Log[List[x]]

which gives a desirable result that splits the expression into pieces. Note that PDF[NormalDistribution[m, s], x] was evaluated of the form Times[a,b,c] and thus u was identified as Times.

However, if I replace u_ by Times, i.e.,

Log[PDF[NormalDistribution[m, s], x]] /. Log[Times[x__] ] :> Plus @@ Log[List[x]]

then, I got an unexpected result that does not change the expression. Can you explain why I should not use Times even though I know that u_ will match Times in the first example?

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    $\begingroup$ Take a look at HoldPattern. Times[x__] is evaluated. $\endgroup$
    – Kuba
    Commented Jun 5, 2015 at 11:52
  • $\begingroup$ @Kuba I don't think I understand you. Could you be more specific? $\endgroup$
    – Sungmin
    Commented Jun 5, 2015 at 11:57

1 Answer 1

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@Kuba is actually pretty specific. 

In[1]:= Log[PDF[NormalDistribution[m, s], x]] /. 
 HoldPattern[Log[Times[x__]]] :> Plus @@ Log[List[x]]

Out[1]= Log[E^(-((-m + x)^2/(2 s^2)))] - 1/2 Log[2 \[Pi]] + Log[1/s]

BTW, you may also interested in PowerExpand.

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