Jeremiah 32:1 states "in the tenth year of Zedekiah king of Judah" How do I find out what year that was?
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If you right-click on "Nebuchadnezzar" in that verse you can select him as a Person and then click Factbook
This open a Factbook entry which includes some timeline information - including when he started to reign
Clicking on that entry opens the timeline shows more detailed information (including when Zedekiah started reigning as well)
There may be other/better ways to do this depending on what resources you have in your library, but using the Timeline tool will give it to you.
Open the timeline, delete the current date range if one is showing, to unclutter the timeline, select the "Type" drop-down menu and uncheck everything except "individual", then in the search window to the right of the "Type" drop-down menu, type zedekiah. Doing this brings up three entries in my Timeline window. The first is for King Zedekiah and shows his period of reign as 597-586 BC.
Ooops! Graham's response wasn't there when I gave the timeline thing a try. Go with Graham! [:)]
You already received great answers which show the power of Logos to research this kind of stuff. If one was just wondering quickly or was asked this question in a group session, I'd probably just look into the notes of any Study Bible resource on this verse (such as Logos' own free FSB, the NET bible notes or the CSSB) and get 587/88 BC.
Another way which will give more information is to open a bible dictionary (AYBD, e.g.) and look up "Zedekaiah"—it's Zedekiah #4 in the AYBD.