       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 -
>    - 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
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 - 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

```

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