Function: log
Section: transcendental
C-Name: glog
Prototype: Gp
Help: log(x): natural logarithm of x.
Description:
 (gen):gen:prec        glog($1, $prec)
Doc: principal branch of the natural logarithm of
 $x \in \C^{*}$, i.e.~such that $\Im(\log(x))\in{} ]-\pi,\pi]$.
 The branch cut lies
 along the negative real axis, continuous with quadrant 2, i.e.~such that
 $\lim_{b\to 0^{+}} \log (a+bi) = \log a$ for $a \in\R^{*}$.
 The result is complex
 (with imaginary part equal to $\pi$) if $x\in \R$ and $x < 0$. In general,
 the algorithm uses the formula
 $$\log(x) \approx {\pi\over 2\text{agm}(1, 4/s)} - m \log 2, $$
 if $s = x 2^{m}$ is large enough. (The result is exact to $B$ bits provided
 $s > 2^{B/2}$.) At low accuracies, the series expansion near $1$ is used.

 $p$-adic arguments are also accepted for $x$, with the convention that
 $\log(p)=0$. Hence in particular $\exp(\log(x))/x$ is not in general equal to
 1 but to a $(p-1)$-th root of unity (or $\pm1$ if $p=2$) times a power of $p$.
Variant: For a \typ{PADIC} $x$, the function
 \fun{GEN}{Qp_log}{GEN x} is also available.
