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**Hint:**A function is a relation which describes that there should be only one output for each input, or we can say that a special kind of relation (a set of ordered pairs), which follows a rule that is every \[x\] value must be associated with a \[y\]value.

**Complete step-by-step solution:**

We know that, \[\ln \left( x \right)\]is defined for all places where \[x > 0\], that is \[x\] should always be positive.

Here, in this question, \[f\left( x \right) = \ln \left( x \right)\] (fractional part of \[x\]). We also know that the range of the fractional part of \[x\] is \[0 \leqslant (x) < 1\].

But, to define \[f\left( x \right)\]as a fractional part of \[x\], \[(x) \ne 0\] , and we also know that \[0 \leqslant (x) < 1\] means that domain is where all the real numbers were \[(x) = 0\].

And \[(x) = 0\]when \[x\] is some kind of an integer.

Therefore, the total set of all the integers numbers must be removed from real numbers.

So, the domain comes out to be \[x = R - Z\].

**So, according to the solution, Option B is the right option.**

**Note:**One thing which we should keep in mind is that \[\ln \left( x \right)\] function is defined for all \[x > 0\]( \[x\] is always negative, not even equal to 0). The Range of the fractional part of x is always \[0 \leqslant (x) < 1\]. In Mathematics, domain is a collection of the first values in the order, and range is the collection of the second values.

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