the graph of log x is given below :-
So, log x is defined for x > 0
So, according to question ,
log(logsinx) will be defined when log(sinx) > 0
So, log(sinx) >0
So, sinx > e^{0}
sinx > 1
Since , we know that -1<=sinx <=1
but we got sinx >1
So, we can say no 'x' is existed here.