Log function and documentation revamp

[rwst]

From https://groups.google.com/forum/?hl=en#!topic/sage-devel/hwm-7V8S3hE

sage: log(int(8),2)
log(8)/log(2)
sage: log(8,int(2))
log(8)/log(2)
sage: log(8,2)
3

Also:
sage: log(1/8,2)
log(1/8)/log(2)

sage: ((-1)^(-I*log(3)/pi)).n()
3.00000000000000
sage: log(3,-1)
...
ValueError: m must be positive

Depends on #21623
Depends on #21517
Depends on #21518

Component: symbolics

Author: Ralf Stephan

Branch/Commit: b0486c8

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/18970

[rwst]

Description changed:

--- 
+++ 
@@ -7,5 +7,9 @@
 log(8)/log(2)
 sage: log(8,2)
 3
+
+Also:
+sage: log(1/8,2)
+log(1/8)/log(2)
 ```
 

[rwst]

Branch: u/rws/always_simplify_log_a_m_a__to_m_for_any_a_m_coercible_to_integer

[rwst]

Author: Ralf Stephan

[rwst]

New commits:

0086ec0

18970: always simplify log(a^m,a) if possible

[rwst]

Commit: 0086ec0

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

92d9762

18970: fix

[sagetrac-git]

Changed commit from 0086ec0 to 92d9762

[dkrenn]

comment:6

Before this patch

sage: %timeit log(QQ(8),2)
The slowest run took 5.08 times longer than the fastest. This could mean that an intermediate result is being cached 
10000 loops, best of 3: 39.4 µs per loop
sage: %timeit log(QQ(8),3)
10000 loops, best of 3: 38.1 µs per loop
sage: %timeit log(ZZ(8),3)
10000 loops, best of 3: 32.8 µs per loop
sage: %timeit log(ZZ(8),2)
The slowest run took 14.52 times longer than the fastest. This could mean that an intermediate result is being cached 
1000000 loops, best of 3: 1.12 µs per loop

After

sage: %timeit log(QQ(8),3)
The slowest run took 37.28 times longer than the fastest. This could mean that an intermediate result is being cached 
10000 loops, best of 3: 48.8 µs per loop
sage: %timeit log(QQ(8),2)
The slowest run took 8.69 times longer than the fastest. This could mean that an intermediate result is being cached 
100000 loops, best of 3: 5.43 µs per loop
sage: %timeit log(ZZ(8),3)
10000 loops, best of 3: 48.4 µs per loop
sage: %timeit log(ZZ(8),2)
The slowest run took 8.80 times longer than the fastest. This could mean that an intermediate result is being cached 
100000 loops, best of 3: 5.34 µs per loop

I've repeated (differently) it due to the caching-warnings. Before

sage: timeit('log(QQ(8),3)', number=1, repeat=1)
1 loops, best of 1: 191 µs per loop
sage: timeit('log(QQ(8),2)', number=1, repeat=1)
1 loops, best of 1: 150 µs per loop
sage: timeit('log(ZZ(8),3)', number=1, repeat=1)
1 loops, best of 1: 125 µs per loop
sage: timeit('log(ZZ(8),2)', number=1, repeat=1)
1 loops, best of 1: 24.1 µs per loop

After

sage: timeit('log(QQ(8),3)', number=1, repeat=1)
1 loops, best of 1: 1.93 ms per loop
sage: timeit('log(QQ(8),2)', number=1, repeat=1)
1 loops, best of 1: 45.1 µs per loop
sage: timeit('log(ZZ(8),3)', number=1, repeat=1)
1 loops, best of 1: 158 µs per loop
sage: timeit('log(ZZ(8),2)', number=1, repeat=1)
1 loops, best of 1: 45.1 µs per loop

In particular log(QQ(8),3) becomes much slower (and only log(QQ(8),2) gains speed).