A) \[-\log ({{\log }_{b}}a)\]
B) \[-{{\log }_{a}}({{\log }_{a}}b)\]
C) \[{{\log }_{a}}({{\log }_{e}}a)-{{\log }_{a}}({{\log }_{e}}b)\]
D) \[{{\log }_{a}}({{\log }_{e}}b)-{{\log }_{a}}({{\log }_{e}}a)\]
Correct Answer: C
Solution :
Obviously \[{{({{a}^{x/2}})}^{2}}={{\log }_{x}}a\ .\ {{\log }_{b}}x={{\log }_{b}}a\] \[\Rightarrow \]\[{{a}^{x}}={{\log }_{b}}a\]\[\Rightarrow \]\[x={{\log }_{a}}({{\log }_{b}}a)\] \[\Rightarrow \]\[x={{\log }_{a}}\left( \frac{{{\log }_{e}}a}{{{\log }_{e}}b} \right)={{\log }_{a}}({{\log }_{e}}a)-{{\log }_{a}}({{\log }_{e}}b)\].
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