As in the complex case, it has an inverse function, named the p-adic logarithm.
The usual exponential function on C is defined by the infinite series Entirely analogously, one defines the exponential function on Cp, the completion of the algebraic closure of Qp, by However, unlike exp which converges on all of C, expp only converges on the disc 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.
[1] The power series converges for x in Cp satisfying |x|p < 1 and so defines the p-adic logarithm function logp(z) for |z − 1|p < 1 satisfying the usual property logp(zw) = logpz + logpw.
[b] This function on C ×p is sometimes called the Iwasawa logarithm to emphasize the choice of logp(p) = 0.
In fact, there is an extension of the logarithm from |z − 1|p < 1 to all of C ×p for each choice of logp(p) in Cp.
Similarly if z and w are nonzero elements of Cp then logp(zw) = logpz + logpw.
Another major difference to the situation in C is that the domain of convergence of expp is much smaller than that of logp.