Learn French: Je suis soulagé de trouver l'auberge après la randonnée. - I am relieved to find the hostel after the hike.

The phrase être soulagé (to be relieved) expresses an emotion, which usually triggers the subjunctive. However, because the person who is relieved and the person finding the hostel are the same (je), we must use the infinitive (trouver) instead. If the subjects were different, you would use the subjunctive: Je suis soulagé que tu trouves l'auberge (I am relieved that you find the hostel).

Why is the word de needed before trouver?

In French, adjectives expressing emotion or a state of mind (like soulagé, content, heureux, or triste) require the preposition de when they are followed directly by an infinitive verb.

If a woman says this, does the spelling of soulagé change?

Yes. Because soulagé (relieved) is an adjective, it must agree with the subject. A woman would write Je suis soulagée, adding an e at the end. However, it sounds exactly the same when spoken.

Does auberge just mean a youth hostel?

It can, but une auberge traditionally means an inn or a country guesthouse. If you specifically want to talk about a youth hostel, the full term is une auberge de jeunesse.