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

• To: mathgroup at smc.vnet.net
• Subject: [mg81697] Re: [mg81671] Simplifying Log[a] + Log[expr_] - Log[2 expr_]: Brute force necessary?
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Tue, 2 Oct 2007 05:25:15 -0400 (EDT)
• References: <200710010846.EAA22900@smc.vnet.net>

On 1 Oct 2007, at 17:46, W. Craig Carter wrote:

>
> Hello,
> 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] -
>    - 2 Log[-2 ((R + x)^2 + y^2 + (z - zvar)^2)]
>     +    2 Log[(R + x)^2 + y^2 + (z - zvar)^2]),
>   Assumptions ->
> {Element[zvar,Reals], Element[x,Reals],Element[y, Reals], Element 
> [z, Reals}]
>
> doesn't simplify. I can't see a way to do this, but brute force.
>
> Any ideas?
> Thanks,
>
> W. Craig Carter
>
>


There are several excellent reason for that. Firstly, there are  
syntax errors in your formula. Secondly, there is no assumption about  
R. But most imortandly, the expression under the first Log sign is  
non-positive hence the formula you wish to apply does not hold.  
Compare it with:

  Simplify[Log[4] - 2 Log[2 ((R + x)^2 + y^2 + (z - zvar)^2)] +
   2 Log[(R + x)^2 + y^2 + (z - zvar)^2], Element[R | x | y | z |  
zvar, Reals]]
  0

Note also that Mathemaica will no collect logs in expressions like:

  Simplify[Log[a] + 2*Log[b], a > 0 && b > 0]
  Log[a] + 2*Log[b]

even though Log[a b^2] has a smaller LeafCount than Log[a]+ 2 Log[b].  
If you want this kind of transformation to be used you have to append  
suitable transformations Simplify using the option  
TransformationFunctions.
Andrzej Kozlowski

• References:
  • Simplifying Log[a] + Log[expr_] - Log[2 expr_]: Brute force necessary?
    • From: "W. Craig Carter" <ccarter@mit.edu>