Learn French: Nous avons un gros budget pour louer un logement. - We have a large budget to rent an accommodation.

In French, when talking about a large amount of money or a substantial fund, it is common to use gros (literally 'fat' or 'thick') or important rather than grand. So, a big budget is naturally un gros budget.

Why is there a pour before louer? The English translation just says 'to rent'.

Whenever the English 'to' actually means 'in order to' or expresses a specific purpose, French uses pour followed by an infinitive verb. Since the budget exists specifically in order to rent, we say pour louer.

Does louer mean 'to rent' or 'to rent out'?

It actually means both! Louer can be used for renting a place to live in as a tenant, or renting out a property you own as a landlord. You have to rely on context to tell them apart. Here, needing a budget implies they are the ones paying to stay somewhere.

What exactly counts as un logement?

Un logement is a very general term for any place where someone lives or stays, such as a house, an apartment, or even a student room. It comes from the verb loger (to accommodate or to house) and translates perfectly to 'accommodation' or 'housing'.