How to find the range of Log_{[x-1]}Sinx, In the base we have [x-1], where [.] means greatest integer function
You may not understand how very strange this question actually is.
If we assume that the domain us a subset of the real numbers then:
$\large\log_b(t)=s$ means that $\large b>0$, $\large t>0$, & $\large b^s=t$.
Therefore, $\large{\log_b(t)=\dfrac{\log(t)}{\log(b)}}$
So think about what you are asking!