the domain of the function f(x) = log[x] base to |x|

Answer 1

If the base is x, then all you need is x > 0
The graph is interesting. See

https://www.wolframalpha.com/input/?i=log_x%28floor%28x%29%29

Answer 2

Oops - my bad. Since [x]=0 for x<1, you need x > 1