Measuring Fractal Dimension ?

[Lie Ryan]

Mark Dickinson wrote:
> On Jun 23, 3:52 am, Steven D'Aprano
> <ste... at REMOVE.THIS.cybersource.com.au> wrote:
>> On Mon, 22 Jun 2009 13:43:19 -0500, David C. Ullrich wrote:
>>> In my universe the standard definition of "log" is different froim what
>>> log means in a calculus class
>> Now I'm curious what the difference is.
> 
> It's just the usual argument about whether 'log' means
> log base 10 or log base e (natural log).  At least in the
> US, most[*] calculus texts (and also most calculators),
> for reasons best known to themselves, use 'ln' to mean
> natural log and 'log' to mean log base 10.  But most
> mathematicians use 'log' to mean natural log:  pick up a
> random pure mathematics research paper that has the word
> 'log' in it, and unless it's otherwise qualified, it's
> safe to assume that it means log base e.  (Except in the
> context of algorithmic complexity, where it might well
> mean log base 2 instead...)

I usually use log without explicit base only when the base isn't
relevant in the context (i.e. when whatever sane base you put in it
wouldn't really affect the operations). In algorithmic complexity, a
logarithm's base doesn't affect the growth shape and, like constant
multiplier, is considered irrelevant to the complexity.

> Python also suffers a bit from this confusion:  the
> Decimal class defines methods 'ln' and 'log10', while
> the math module and cmath modules define 'log' and
> 'log10'.  

In fact, in the Decimal class there is no log to an arbitrary base.

> (But the Decimal module has other problems,
> like claiming that 0**0 is undefined while
> infinity**0 is 1.)

Well, in math inf**0 is undefined. Since python is programming language,
and in language standards it is well accepted that undefined behavior
means implementations can do anything they like including returning 0,
1, 42, or even spitting errors, that doesn't make python non-conforming
implementation.

A more serious argument: in IEEE 745 float, inf**0 is 1. Mathematic
operation in python is mostly a wrapper for the underlying C library's
sense of math.

> [*] A notable exception is Michael Spivak's 'Calculus', which also
> happens to be the book I learnt calculus from many years ago.