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However, unlike exp which converges on all of '''C''', exp<sub>''p''</sub> only converges on the disc
:<math>|z|_p<p^{-1/(p-1)}.</math>
This is because ''p''-adic series converge [[if and only if]] the summands tend to zero, and since the ''n''! in the denominator of each summand tends to make them large ''p''-adically, a small value of ''z'' is needed in the numerator. It follows from [[Legendre's formula]] that if <math>|z|_p < p^{-1/(p-1)}</math> then <math>\frac{z^n}{n!}</math> tends to <math>0</math>, ''p''-adically.
Although the ''p''-adic exponential is sometimes denoted ''e''<sup>''x''</sup>, the [[e (mathematical constant)|number ''e'']] itself has no ''p''-adic analogue. This is because the power series exp<sub>''p''</sub>(''x'') does not converge at {{nowrap|''x'' {{=}} 1}}. It is possible to choose a number ''e'' to be a ''p''-th root of exp<sub>''p''</sub>(''p'') for {{nowrap|''p'' ≠ 2}},{{efn|or a 4th root of exp<sub>2</sub>(4), for {{nowrap|''p'' {{=}} 2}}}} but there are multiple such roots and there is no canonical choice among them.<ref>{{harvnb|Robert|2000|p=252}}</ref>
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The power series
:<math>\log_p(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}x^n}{n},</math>
converges for ''x'' in '''C'''<sub>''p''</sub> satisfying |''x''|<sub>''p''</sub> < 1 and so defines the '''''p''-adic logarithm function''' log<sub>''p''</sub>(''z'') for |''z'' − 1|<sub>''p''</sub> < 1 satisfying the usual property log<sub>''p''</sub>(''zw'') = log<sub>''p''</sub>''z'' + log<sub>''p''</sub>''w''. The function log<sub>''p''</sub> can be extended to all of {{SubSup|'''C'''|''p''|×}} (the set of nonzero elements of '''C'''<sub>''p''</sub>) by imposing that it continues to satisfy this last property and setting log<sub>''p''</sub>(''p'') = 0. Specifically, every element ''w'' of {{SubSup|'''C'''|''p''|×}} can be written as ''w'' = ''p<sup>r</sup>''·ζ·''z'' with ''r'' a [[rational number]], ζ a [[root of unity]], and |''z'' − 1|<sub>''p''</sub> < 1,<ref>{{harvnb|Cohen|2007|loc=Proposition 4.4.44}}</ref> in which case log<sub>''p''</sub>(''w'') = log<sub>''p''</sub>(''z'').{{efn|In factoring ''w'' as above, there is a choice of a root involved in writing ''p<sup>r</sup>'' since ''r'' is rational; however, different choices differ only by multiplication by a root of unity, which gets absorbed into the factor ζ.}} This function on {{SubSup|'''C'''|''p''|×}} is sometimes called the '''Iwasawa logarithm''' to emphasize the choice of log<sub>''p''</sub>(''p'') = 0. In fact, there is an extension of the logarithm from |''z'' − 1|<sub>''p''</sub> < 1 to all of {{SubSup|'''C'''|''p''|×}} for each choice of log<sub>''p''</sub>(''p'') in '''C'''<sub>''p''</sub>.<ref>{{harvnb|Cohen|2007|loc=§4.4.11}}</ref>
==Properties==
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