When writing a formal letter, what is the correct format for the date?
Aŭgusto 5, 2019?
5 aŭgusto 2019?
5 Aŭgusto 2019?
2019 aŭgusto 5?
2019 Aŭgusto 5?
(when do you capitalize the month's name?)
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If you have to (or want to) spell the month name out, the usual date format in both written and spoken Esperanto would be
la 5-a de aŭgusto 2019
With the numbers spelled out, it'd be
la kvina de aŭgusto du mil dek naŭ
which is of course also how the above version would be read aloud or pronounced in spoken texts.
Note that if the date appears in prose (e.g. as part of a sentence), you can't drop the definite article la. Also, the ordinal numeral (i.e., with the -a adjective ending) must be used instead of a bare number (cardinal numeral) for any day of a month, not just the first as sometimes in written English. I assume that in non-prose contexts (e.g. a list of dates or a table) you may leave out the definite article la, but not the ordinal ending -a.
If you feel that that's too long or cumbersome, consider whether a purely numerical date would suffice. Usually, this should be the case in any letter, whether formal or not. If so, in any international context no matter the language (thus definitely in Esperanto, too) you should choose the "long" ISO 8601 format:
(The "short" ISO 8601 format 20190805 can be confused with big numbers. All other (non-ISO) date formats tend to be both country- or language-specific and allow misinterpretation in other languages or countries.)
(Comic CC BY-NC 2.5 xkcd.com)