
ausrufen
to exclaim
To call out something loudly and often emotionally so that others can hear it; to make an exclamation.
„Endlich!“, rief er aus, als der Zug im Bahnhof einfuhr.
"Finally!" he exclaimed as the train pulled into the station.
to exclaim / proclaim
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to exclaim
To call out something loudly and often emotionally so that others can hear it; to make an exclamation.
„Endlich!“, rief er aus, als der Zug im Bahnhof einfuhr.
"Finally!" he exclaimed as the train pulled into the station.

to proclaim
To announce or proclaim something officially and publicly, for example a state of affairs, a competition, or a winner.
Die Regierung hat den nationalen Notstand ausgerufen.
The government has proclaimed a national state of emergency.
'Ausrufen' (to exclaim / proclaim) is a separable verb. The prefix 'aus-' goes at the end in a main clause. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Der Schiedsrichter rief den Sieger laut aus.
The referee loudly proclaimed the winner.
In the meaning 'etwas laut rufen' (to call something out loudly), 'ausrufen' (to exclaim) is often used to introduce direct speech. The reporting clause can come before or after the quotation.
„Ich hab's geschafft!“, rief sie glücklich aus.
"I did it!" she exclaimed happily.
In the sense of 'offiziell verkünden' (to announce officially), 'ausrufen' (to proclaim) is often used with an accusative object that describes a state or a result. Typical examples are: 'den Notstand ausrufen' (to proclaim a state of emergency), 'den Sieger ausrufen' (to announce the winner), 'einen Wettbewerb ausrufen' (to announce a competition), or 'die Republik ausrufen' (to proclaim the republic).
Nach dem Erdbeben rief die Regierung den Notstand aus.
After the earthquake, the government proclaimed a state of emergency.
You can tell from the context whether 'ausrufen' (to exclaim / proclaim) means an emotional exclamation or an official announcement. Personal, emotional statements point to the first meaning, while formal, public announcements point to the second meaning.
„Vorsicht!“, rief er schnell aus.
"Watch out!" he quickly exclaimed.
to explain / declare
Erklären and ausrufen can both refer to an official proclamation, but they are used in different fixed expressions. You erklären (declare) a war, but ausrufen (proclaim) a state of emergency. The choice therefore depends on which status is being officially put into effect.

Nach dem schweren Erdbeben ruft die Regierung den Notstand aus.
After the severe earthquake, the government proclaims a state of emergency.
Nach dem Angriff erklärt das Land seinem Nachbarn den Krieg.
After the attack, the country declares war on its neighbor.