The English word Log will be understood, as explained in the other answer. It is related (and obviously similar) to the German word Logbuch, which denotes logs or diaries aboard ships and some other vehicles (such as aircraft or spaceships), as well as some logs in contexts that might be metaphorically related to the aforementioned ship context1. However, Log itself does not seem to have been used in the sense of the English log in German before being introduced via IT contexts; Google NGram shows a certain rise in frequency of the term Log in German texts starting between 1990 and 2000, and at least random samples look like after 2000, quite some books mention Log related to software logs, whereas before 1990, books seem to almost exclusively refer to logarithms when they write Log.
However, the actual German translation of log would be Protokoll. It is a rather frequently used word that appears in many contexts in German (Protokolle are created on business meetings, on meetings between parents and schoolteachers, on organized meetings (to make decisions) between tenants of a house, during court trials, etc.), so every native speaker will likely know it. Some examples of real-world IT-related usage:
Based upon that word, the expressions you asked about are:
- A log: ein Protokoll
- Logging to a file: [den Ablauf/die Ausführung/…] in einer Datei protokollieren
- A log entry: ein Protokolleintrag; ein Eintrag/eine Zeile/… im Protokoll
Concerning the expression with booking, I am not quite sure — does that mean you call the act of logging something “booking an entry into a log”? In that case, I would still use protokollieren in German.
Alternatively, do you mean that a booking (of something else) is getting logged? In that case, log booking seems like a misnomer to start with, as it should be either a logged booking, or a booking [in a] log, i.e. something like Buchungsprotokoll (if the whole log only contains bookings), or protokollierte Buchung (for a single event).
1: This contextual restriction of where a Logbuch would be used is also illustrated in older text samples provided by userunknown:
- Reise nach dem Vorgebirge der guten Hoffnung, ..., Andreas Sparrmann, Berlin 1784: "Da ausser mir nur sehr wenige Personen von der Wache auf dem Verdeck waren, wurde dieser Vorfall weder in das Logbuch, noch in eins von den bereits gedruckten Tagebüchern eingetragen" (the protagonist is travelling on a ship, hence there is a Logbuch)
- Reisen eines Officiers durch die Schweiz und Italien, Hannover, 1786, "Er schreibet auch auf eine schwarze Tafel den Lauf, Wind, und die Anmerkungen, welches alle 24 Stunden ins Logbuch oder Schifsjüurnal getragen wird." (a letter describing the operations aboard a ship)
- Reisebeschr. nach Arabien u a umliegenden Ländern, Carsten Niebuhr, 1774, "Da ich bemerkte, daß unsere Rechnung nach dem Logbuch uns bey unsere Ausreise weiter nördlich und bey unserer Zurückkunft weiter südlich brachte als wir wirklich waren (...)" (this, again, describes a journey aboard a ship)
(Of course, this does not mean the word can absolutely never appear in another context; almost every word can be used in any context and meaning, given the appropriate metaphor. However, such occurrences do not seem to be relevant when looking at the typical usage of a word.)