A) \[({{\log }_{e}}3)({{\log }_{e}}2)\]
B) \[{{\log }_{e}}3\]
C) \[{{\log }_{e}}2\]
D) \[\frac{{{\log }_{e}}5}{{{\log }_{e}}3}\]
E) \[\frac{{{\log }_{e}}3}{{{\log }_{e}}2}\]
Correct Answer: A
Solution :
\[{{\log }_{e}}3-\frac{{{\log }_{e}}9}{{{2}^{2}}}+\frac{{{\log }_{e}}27}{{{3}^{2}}}-\frac{{{\log }_{e}}81}{{{4}^{2}}}....\] \[={{\log }_{e}}3\left( 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}..... \right)\] \[={{\log }_{e}}3{{\log }_{e}}2\]
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