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Find the range of f(x)=(log)([x-1])sinxd...

Find the range of `f(x)=(log)_([x-1])sinxdot`

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The correct Answer is:
`(-oo,0]`
`f(x)= log_[x-1] sinx` where [.] denotes the greatest integer.
To get the range fo `f(x)`, let us examine the values of x for which the function is defined.
`f(x)` is defined if `sinx gt 0 " and " [x-1] gt 0 " and " [x-1] ne 1,` i.e.,
`0 lt sinx le 1 " and " [x] gt 2`
Now, for base of the logarithm is more than '2' and `sinx in (0,1],` clearly, `log_([x-1]) sinx in (-oo,0].`
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