Question
The domain of the real-valued function $$f\left( x \right) = {\log _e}\left| {{{\log }_e}x} \right|$$ is :
A.
$$\left( {1,\, + \infty } \right)$$
B.
$$\left( {0,\, + \infty } \right)$$
C.
$$\left( {e,\, + \infty } \right)$$
D.
none of these
Answer :
none of these
Solution :
$$\eqalign{
& x > 0{\text{ and }}{\log _e}x \ne 0 \cr
& \therefore x > 0{\text{ and }}x \ne 1 \cr
& \therefore {\text{domain}} = \left( {0,\,1} \right) \cup \left( {1,\, + \infty } \right) \cr} $$