A) \[1-lo{{g}_{e}}2\]
B) \[1+lo{{g}_{e}}2\]
C) \[lo{{g}_{e}}2\]
D) \[lo{{g}_{e}}3\]
E) \[1+lo{{g}_{e}}3\]
Correct Answer: C
Solution :
Let\[S={{\log }_{9}}3+{{\log }_{27}}3-{{\log }_{81}}3+{{\log }_{243}}3-....\] \[={{\log }_{{{3}^{2}}}}3+{{\log }_{{{3}^{3}}}}3-{{\log }_{{{3}^{4}}}}3+{{\log }_{{{3}^{5}}}}3-....\] \[=\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-....+1-1\] \[=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-....\] \[={{\log }_{e}}(1+1)={{\log }_{e}}2\]
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