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From Elxis Official Documentation

Logo Multi Language (v.2.0) it is a module for Elxis 2008.x - 2009.x that displays a different Logo for any published language of your site. The logo is link to your home page of any language and has as title and alternate text (on mouse over) the site name.

Usage: Create a folder with name logos into folder images and load your logos for any language naming as logo_language.xyz where xyz is the file type of logo image which you select from module parameters and can be gif or jpg or png (example: logo_greek.jpg for Greek, logo_english.jpg for English, logo_french.jpg for French e.t.c.). Logo images must be of the same type.

Note: If a logo image of a published language does not exist, then will be displayed the english one which is the default.

So when you create a css template, on the top header where usually is set logo, create a position for the logo as following:

1. Into file layout.css of your template create an id for logo

#sitelogo {
  float: left;
  margin-top: 0;
}

2. Into file index.php of your template set logo id

<div id="topheader">
<div id="sitelogo">
<?php elxLoadModule('mod_logo', -2); ?>
</div>
</div>

3. Into Elxis administration console set module Logo Multi Language in position hidden published in all languages.

And is done!