base is greater than 0 and not equal to 1.
So
[x-1] > 0 and [x-1] not equal to 1
so x >= 2 and x not equal to 2
so the intersection is:
x > 2

log_[x-1]sin(x)  = y

[x-1] ^y = sin x

now as x>2

so [x-1] will vary from (1, infinity) .. 1 is not included.

so for the least value of [x-1] .. which is. tending to 1 from right hand side.

y will be 0, as sinx can have values between -1 and 1.

similarly for the highest value of [x-1] which is infinity, the power y will tend to -infinity to make the net value 1. ( as[x-1]'s value increases, the power increases in the negative)

so the range (y) = [0, -infinity)