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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