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

• To: mathgroup at smc.vnet.net
• Subject: [mg81803] Re: Simplifying Log[a] + Log[expr_] - Log[2 expr_]: Brute force necessary?
• From: Peter Breitfeld <phbrf at t-online.de>
• Date: Wed, 3 Oct 2007 06:33:14 -0400 (EDT)
• References: <fdqclq$mmg$1@smc.vnet.net>

W. Craig Carter schrieb:
>
> 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
>

You can use a rule to bring everything under one Log:

LogZusammenRule={
  n_. Log[a_]+m_. Log[b_]:>Log[a^n b^m],
  n_. Log[a_]-m_. Log[b_]:>Log[a^n/b^m],
  a_ Log[b_]:>Log[b^a] };

Then your expression  

ll= - 2 Log[-2 ((R + x)^2 + y^2 + (z - zvar)^2)] +
       2 Log[(R + x)^2 + y^2 + (z - zvar)^2]

will be reduced to Log[4]:

ll/.LogZusammenRule   =====>  Log[4]
	   
Gruss Peter
-- 
==-==-==-==-==-==-==-==-==-==-==-==-==-==-==-==-==-==-==-==-==-==
Peter Breitfeld, Bad Saulgau, Germany -- http://www.pBreitfeld.de

• Follow-Ups:
  • Re: Re: Re: Simplifying Log[a] + Log[expr_] -
    • From: "W. Craig Carter" <ccarter@mit.edu>
  • Re: Re: Simplifying Log[a] + Log[expr_] - Log[2 expr_]: Brute force necessary?
    • From: Andrzej Kozlowski <akoz@mimuw.edu.pl>