Q. The sum of the series $\log_{9}3 + \log_{27}3 - \log_{81}3 + \log_{243}3 - ...$ is
A) $1 - \log_{e}2$
B) $1 + \log_{e}2$
C) $\log_{e}3$
D) $1 + \log_{e}3$
Explanation
(c)
The series is $\log_{3^{2}}3 + \log_{3^{3}}3 - \log_{3^{4}}3 + \log_{3^{5}}3 - ...$
$= \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - ... + 1 - 1$
$= 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - ...$
$= \log_{e}{(1 + 1)} = \log_{e}2$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.