# Logarithmic Series

Expansion of log

*(1 +*_{e}*x*); if |*x*| < 1, thenlog

*(1 +*_{e}*x*) =*x*â€“Replacing

*x*by â€“*x*in the logarithmic series, we getlog

*(1 â€“*_{e}*x*) = â€“*x*â€“or â€“ log

*(1 â€“*_{e}*x*) =*x*+# Some important results from logarithmic series

- i. log
(1 +_{e}*x*) + log(1 â€“_{e}*x*) = log(1 â€“_{e}*x*^{2})*x*< 1)

ii. log(1 +_{e}*x*) â€“ log(1 â€“_{e}*x*)=_{e} - The series expansion of log
(1 +_{e}*x*) may fail to be valid, if |*x*| is not less than 1. It can be proved that the logarithmic series is valid for*x*= 1. Putting*x*= 1 in the logarithmic series, we get2 = 1 â€“ + â€¦ âˆž_{e}