
lautstark
loud
With great volume; very loud. Describes noises, voices, or music.
Die Fans feierten den Sieg lautstark in der Innenstadt.
The fans celebrated the victory loudly in the city center.
loud / vociferous
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loud
With great volume; very loud. Describes noises, voices, or music.
Die Fans feierten den Sieg lautstark in der Innenstadt.
The fans celebrated the victory loudly in the city center.

vociferous
Expressed emphatically and unmistakably, often in connection with protest, criticism, or demands.
Die Opposition protestierte lautstark gegen das neue Gesetz.
The opposition vociferously protested against the new law.
'Lautstark' is a compound adjective made up of 'laut' (loud) and 'stark' (strong). So it literally means something like 'strongly loud' and intensifies the quality of being loud.
Die Kinder spielten lautstark im Garten.
The children played loudly in the garden.
'Lautstark' (loudly/vociferously) is very often used as an adverb to describe a verb. It then remains unchanged in the sentence and describes how an action is carried out.
Die Nachbarn haben bis spät in die Nacht lautstark Musik gehört.
The neighbors listened to music loudly until late into the night.
In the meaning of 'emphatically', 'lautstark' (vociferously) is often combined with verbs of demanding or criticizing. Typical phrases are 'lautstark protestieren' (to protest vociferously), 'lautstark fordern' (to demand vociferously), and 'lautstark kritisieren' (to criticize vociferously).
Die Bürgerinitiative forderte lautstark mehr Lärmschutzmaßnahmen.
The citizens' initiative vociferously demanded more noise-protection measures.
vehement
Vehement and lautstark can describe strong criticism or demands. Vehement primarily emphasizes the force and passion of the delivery, whereas lautstark emphasizes that the statement is expressed publicly and cannot be ignored. Choose vehement for intensity and lautstark for the audible effect.

Die Anwohner protestierten lautstark vor dem Rathaus.
The residents protested vociferously in front of city hall.
Die Anwohner widersprachen dem Bauprojekt vehement.
The residents vehemently objected to the construction project.