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Question

The domain of f(x)=logx{x} is
({x} represents the fractional part of x)

A
R
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B
[0,1]
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C
(0,1)
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D
(0,1]
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Solution

The correct option is C (0,1)
For f to be defined,
logx{x}0
and x1,{x}>0 (1)

We know that logax is strictly decreasing for 0<a<1 and strictly increasing for a>1
So, if 0<x<1,
then logx{x}0{x}x0
i.e., {x}1
But 0{x}<1 (2)
From (1) and (2),
{x}(0,1)0<x<1

If x>1,
then {x}1, which is not possible.

Hence, D(f)=(0,1)

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