Question
Let $$f\left( x \right) = {\log _{{x^2}}}25$$ and $$g\left( x \right) = {\log _x}5$$ then $$f\left( x \right) = g\left( x \right)$$ holds for $$x$$ belonging to :
A.
$$R$$
B.
$$\left( {0,\,1} \right) \cup \left( {1,\, + \infty } \right)$$
C.
$$\phi $$
D.
none of these
Answer :
$$\left( {0,\,1} \right) \cup \left( {1,\, + \infty } \right)$$
Solution :
$$\eqalign{
& {\text{Domain of }}f = {D_1} = \left( { - \infty ,\, - 1} \right) \cup \left( { - 1,\,0} \right) \cup \left( {0,\,1} \right) \cup \left( {1,\, + \infty } \right) \cr
& {\text{Domain of }}g = {D_2} = \left( {0,\,1} \right) \cup \left( {1,\, + \infty } \right) \cr
& {\text{So, }}{D_1} \cap {D_2} = \left( {0,\,1} \right) \cup \left( {1,\, + \infty } \right) \cr
& {\text{In this set, }}{\log _{{x^2}}}25 = \frac{1}{2} \times 2\,{\log _x}5 = {\log _x}5 \cr} $$