
laut
loud
With a high volume; clearly audible and not quiet.
Die Musik auf der Party war so laut, dass wir uns kaum unterhalten konnten.
The music at the party was so loud that we could hardly talk to each other.
loud / according to
Loading your saved state.

loud
With a high volume; clearly audible and not quiet.
Die Musik auf der Party war so laut, dass wir uns kaum unterhalten konnten.
The music at the party was so loud that we could hardly talk to each other.

according to
Used as a preposition to indicate a source of information; means 'gemäß' (according to) or 'zufolge' (according to).
Laut dem Wetterbericht soll es morgen den ganzen Tag regnen.
According to the weather report, it's supposed to rain all day tomorrow.
The word 'laut' has two completely different functions. As an adjective, it describes a high volume. As a preposition with the meaning 'gemäß' (according to), it introduces a source of information and comes before a noun.
Die Musik ist mir zu laut.
The music is too loud for me.
When 'laut' is used as a preposition, the following noun can be in the genitive or dative. In modern language, especially in spoken German, the dative is preferred.
Laut dem neuen Gesetz ist das Rauchen hier verboten.
According to the new law, smoking is prohibited here.
Like many adjectives, 'laut' can also be used as an adverb to describe how an action is performed. In this case, it is not declined and usually comes after the verb.
Er lachte laut über den Witz.
He laughed loudly at the joke.
to / after / according to
Nach and laut can attribute information or an assessment to a source. Nach often appears in phrases such as „nach Angaben“ or „nach Einschätzung“ and can also mark a conclusion. Laut names the source more directly, especially in a report or an explicit statement.

Laut der Ärztin ist die Verletzung harmlos.
According to the doctor, the injury is harmless.
Nach Einschätzung der Ärztin ist die Verletzung harmlos.
According to the doctor’s assessment, the injury is harmless.