Log base 1

I claim
$\log_1 n = n$
  (or maybe $n-1$?)
Oops -- this isn't right. $\log_1 n$ diverges for $n\neq 1$.
What I mean is rather the question of encoding information, as noted
below.

Why?
Explanation 1:
Think in terms of unary:
you need a length $\log_b n$ string to encode $n$ in base $b$.
[This is slightly illegit and confusing b/c we're conflating variable
length unary and fixed length $b$-ary, ignoring that length is part of
the data of a variable length string. It's legit b/c the error is size
$\log$, which is small.]
...and you need $n$ digits of unary to encode $n$
(oic: the point is information encoded in a variable length string)
*******
Ok, deeper point is to talk about -digits-;
I've written about this in the "in depth/basics" book
*******

Explanation 2:
$\log_b n = \frac{\log n}{\log b}$
Apply L'H\^ opital's rule, so
$$\log_1 n := \lim_{b\to 1^+} \log_b n = $$
...oops, goes to infinity.

--------------------------------
[This is similar to $0^0 = 1$: think of $n^k = #\Hom(k,n)$.]
BTW, these pathological questions are -not- just silly geeking
questions kids ask -- they were -really asked- by pro mathematicians
of the age (Euler asking in what way $1 - 1 + 1 - \dots ``='' 1/2$),
and clarify what you -mean- when you write "equals" or "limit".

It's also related to "what operations are legit"?
(I believe Taylor? wrote a critique of reckless manipulation of
infintesimals)

This is important:
people may say:
sure, 1 + 1/2 + 1/4 + ... -approaches- 2,
but it never -equals- it.
The retort is:
"In the limit, it -does-:
 you are saying that the -partial sums- are always strictly less,
 which is true, but the limit -is a real number-, and -there is no
 real number just less than 2-.
 However, if you want to talk about hyper-reals, it is possible to
 make sense of `just less than 2' -- but this is much work, and there
 is still the notion of `the underlying real number'"
