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Maliwan weapons often have "Bonus Elemental Damage" listed in their attributes. I'm not entirely sure what to make of this; does it just refer to the listed elemental effect damage, which is presumably higher than it would otherwise be, or does elemental damage from these guns deal even further damage beyond the listed stat due to this effect? Is there an exact amount or percent of additional damage this attribute reflects?

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It's an additional splash damage amount in addition to the weapon card damage. Percentages vary by weapon type.

Some Maliwan weapons list "Deals Bonus Elemental Damage" as an effect. These weapons deal an additional portion of elemental damage as splash damage around the bullet impact. The splash damage equals 50% of the weapon card damage for sniper rifles and Plasma Casters, and 80% for pistols; it is multiplied as normal for the element type, but is not further multiplied by critical hits or badass rank "elemental effect damage." Amp shield damage only applies on direct hits, and not this splash damage. However, the splash damage can trigger an elemental damage over time effect independently of the main bullet, resulting in double DoT triggers on some shots. This effect is especially useful on slag-snipers as the elemental explosion is guaranteed to slag the target even on a non-direct hit, as opposed to non-Maliwan sniper rifles that need to hit the target in addition to having only a reduced chance of applying the slag effect.

Rocket Launchers with "Deals Bonus Elemental Damage" are a special case. Whereas normal rocket launchers simply deal their weapon card damage as splash damage in a certain radius, DBED rocket launchers have two separate splash ranges. The smaller splash range deals 50% of the weapon card damage, and the larger splash range deals 25% of the weapon card damage. These stack, however, so anything within the smaller range is also hit by the larger range, resulting in a total of 75% weapon card damage on close hits. As with pistols and sniper rifles, these two damage effects can trigger the damage over time effect independently.

Source: Borderlands Wiki, "Borderlands 2 Elemental Damage - Bonus Elemental Damage"

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It just refers to the elemental damage already described in the gun stats.

All written effects such as 75% more damage or 20% higher accuracy are already reflected in the written stats.

Added: Slag does not do damage on its own. It is actually very special because it doubles other damage dealt. Which makes one guy with a slag weapon genius in a 4 player co-op session. When playing alone slag weapons are generally not as useful, except against sturdier enemies along with switching guns.

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What about Slag damage then? I've seen Slag guns with the same effect...does it just do nothing for those or is the slag bonus greater for those guns or something? –  Ben Brocka Sep 29 '12 at 20:40
    
Slag does not do damage on its own. It is actually very special because it doubles other damage dealt. Which makes one guy with a slag weapon genius in a 4 player co-op session. When playing alone slag weapons are generally not as useful, except against sturdier enemies along with switching guns. –  Schottlicious Sep 30 '12 at 12:09
    
Yeah, I know that, but I'm wondering what the bonus is supposed to mean in that sense, exactly because slag doesn't deal elemental damage on it's own –  Ben Brocka Sep 30 '12 at 16:11
    
Hmmm, thats actually really interesting. Hadn't thought about that. It seems like it should either affect the chance to slag or the slag damage multiplier. I fear I do not know which. –  Schottlicious Oct 1 '12 at 1:32
    
Slag works wonders as a Single-Player Axton –  Sconibulus Oct 31 '13 at 18:12
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