To walk heavily and slowly through something difficult, or to work hard at something for a long time without seeing quick results.
Possibly from Scandinavian origins, though the exact source is unclear; first appears in English in the 1800s meaning 'to walk heavily through mud.'
Soldiers in World War I used 'slog' constantly to describe marching through muddy trenches, and the word so perfectly captures exhausting progress that it became universal shorthand for any grueling, incremental effort.
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