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Re: Simplifying Log[a] + Log[expr_] - Log[2 expr_]: Brute force necessary?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg81693] Re: Simplifying Log[a] + Log[expr_] - Log[2 expr_]: Brute force necessary?
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Tue, 2 Oct 2007 05:23:11 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, UK
  • References: <fdqclq$mmg$1@smc.vnet.net>

W. Craig Carter wrote:

> This works as I would hope it would:
> 
> Simplify[Log[a^2] + Log[b^2] - Log[-2 b^2],
>   Assumptions -> Element[a, Reals] && Element[b, Reals]]
> 
> It returns -Log[-2/a^2]
> 
> However, something a little more complicated:
> 
> Simplify[
> Log[4] -
---------^
Too many minus signs.
>    - 2 Log[-2 ((R + x)^2 + y^2 + (z - zvar)^2)]
>     +    2 Log[(R + x)^2 + y^2 + (z - zvar)^2]),
------------------------------------------------^
Extraneous parenthesis.
>   Assumptions ->
> {Element[zvar,Reals], Element[x,Reals],Element[y, Reals], Element[z, Reals}]
----------------------------------------------------------------------------^
Missing square bracket.
> 
> doesn't simplify. I can't see a way to do this, but brute force.
> 
> Any ideas?

Fixing the syntax errors and adding the parameter R in the list of real 
argument does not help. You could use *ComplexExpand*.

In[1]:= Simplify[
  Log[4] - 2 Log[-2 ((R + x)^2 + y^2 + (z - zvar)^2)] +
   2 Log[(R + x)^2 + y^2 + (z - zvar)^2],
  Assumptions -> Element[{R, zvar, x, y, z}, Reals]]

Out[1]= Log[4] - 2 Log[-2 ((R + x)^2 + y^2 + (z - zvar)^2)] +
  2 Log[(R + x)^2 + y^2 + (z - zvar)^2]

In[2]:= ComplexExpand[
  Log[4] - 2 Log[-2 ((R + x)^2 + y^2 + (z - zvar)^2)] +
   2 Log[(R + x)^2 + y^2 + (z - zvar)^2]]

Out[2]= -2 \[ImaginaryI] \[Pi] - 2 Log[2] + Log[4]

HTH,
-- 
Jean-Marc


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