▁arguments : ▁ ▁In ▁tropical ▁analysis , ▁this ▁is ▁the ▁sum ▁in ▁the ▁log ▁sem iring . ▁ ▁Properties ▁ ▁The ▁Log Sum Exp ▁function ▁domain ▁is ▁, ▁the ▁real ▁coordinate ▁space , ▁and ▁its ▁range ▁is ▁, ▁the ▁real ▁line . ▁The ▁larger ▁the ▁values ▁of ▁ ▁or ▁their ▁deviation , ▁the ▁better ▁the ▁approximation ▁becomes . ▁The ▁Log Sum Exp ▁function ▁is ▁convex , ▁and ▁is ▁strictly ▁monot on ically ▁increasing ▁everywhere ▁in ▁its ▁domain ▁( but ▁not ▁strictly ▁convex ▁everywhere ). ▁ ▁L SE ▁is ▁a ▁smooth ▁maximum ▁because , ▁applying ▁the ▁tang ent ▁line ▁approximation ▁ ▁if ▁one ▁term , ▁ ▁is ▁much ▁larger ▁than ▁the ▁rest , ▁the ▁second ▁term ▁is ▁small ▁because ▁it ▁has ▁ ▁in ▁the ▁denomin ator , ▁and ▁one ▁gets : ▁ ▁Indeed , ▁there ▁are ▁the ▁following ▁tight ▁bounds ▁( if ▁, ▁otherwise ▁the ▁first ▁inequality ▁is ▁not ▁strict ): ▁ ▁The ▁upper ▁bound ▁is ▁equality ▁if ▁and ▁only ▁if ▁all ▁ ▁are ▁equal . ▁ ▁This ▁is ▁because ▁ ▁( a ▁sum ▁is ▁at ▁most ▁its ▁maximum ▁term ▁each ▁time ), ▁and ▁for ▁positive ▁numbers , ▁ ▁for ▁any ▁term , ▁include ▁the ▁maximum ▁( since ▁it ' s ▁adding ▁positive ▁numbers ), ▁and ▁in ▁fact ▁is ▁strict ▁if ▁ ▁( since ▁you ' re ▁adding ▁a ▁positive ▁number ). ▁Com bin ing ▁with ▁log arith ms ▁and ▁ex ponents , ▁one ▁gets : ▁ ▁The ▁lower ▁bound ▁is ▁met ▁only ▁for ▁, ▁otherwise ▁it ▁is ▁strict ▁but ▁approached ▁when ▁all ▁but