If you slog through something, you work hard and steadily through it.

If you describe a task as a slog, you mean that it is tiring and requires a lot of effort.

If you slog somewhere, you make a long and tiring journey there.

A slog is a long tiring journey.

If two or more people slog it out, they work very hard to try to be the one who is successful or who has their ideas and wishes accepted.