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If $\lfloor{\log_b{n}}\rfloor+1$ counts the number of digits required to represent a number $n$ of radix $b$, what is the significance of $\log_1{n}$ in terms of counting digits of unary numbers?

For instance:

  • $\lfloor\log_{10}{5}\rfloor+1=1$
  • $\lfloor\log_{2}{101_2}\rfloor+1=3$
  • $\lfloor\log_{1}{00000_1}\rfloor+1$ is undefined although you would expect the pattern to continue and tell us that $5$ in base-1 has indeed $5$ digits.

This would effectively mean that $\lfloor\log_1{n}\rfloor+1=n$ and would serve as an identity function. It does not and here's why:

$$\log_b{n}=\frac{\log{n}}{\log{b}}$$ $$\log_1{n}=\frac{\log{n}}{\log{1}}=\frac{\log{n}}{0}$$ and so $\log_1{n}$ for any real $n$ is undefined.

This is further supported by the fact that $\lim_{b\to1^{+}}{\log_b{n}}=\infty$ and $\lim_{b\to1^{-}}{\log_b{n}}=-\infty$ for $n>1$.

Not to mention the very logical contradiction with how any exponential with base of $1$ must equal $1$.

Why does this digit counting property of logarithms make sense but not in the general case? To extend the idea further, is there any other way to logically associate $\log_1{n}$ with the identity function?

bitconfused
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    There is no such thing as log with base 1. – Mariano Suárez-Álvarez Apr 03 '18 at 05:02
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    "and so $\log_1n$ for any real n is undefined"..."Is there any such relationship between $ \log_1n$ and the supposedly reasonable digit counting pattern shown above?" How can it when you (you, yourself) just went to great length to show $\log_1 n$ con not be defined? – fleablood Apr 03 '18 at 05:25
  • Is there any analytical significance to something that is not capable of being defined? Um, no. There is not. My insigt is that without definition this is a logistically void and meaningless and unanswerable question. What I find odd, is that you,the person asking it, are the one who showed it to be baseless and meaningless. That is very unusual. – fleablood Apr 03 '18 at 05:50
  • It is also meaningless to define the infinite summations of divergent series, yet analytically continued Dirichlet series like the Riemann Zeta function logically associate values to them. This isn't uncommon. – bitconfused Apr 03 '18 at 05:56
  • @bitconfused, Sometimes and sometimes not. That Riemannian was able to make sense of something by meromorphically prolonging some function does not mean that it makes sense to talk of random things. If you intend to talk about logarithms in base 1, you first have to make precise exactly what you mean by that —it has no meaning at all normally. That someone did something with something else does not make what you say about logarithms in base 1 make sense. – Mariano Suárez-Álvarez Apr 04 '18 at 01:44

2 Answers2

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Added emphasis on the parts of the original post where the confusion lies...

$\lfloor\log_{1}{00000_1}\rfloor+1$ is undefined although you would expect the pattern to continue and tell us that $5$ in base-1 has indeed $5$ digits.

Calling the unary representation "base-1" is more of a poetic license than math reality, although it is casually (mis)used that way quite often. Indeed, the very article hyperlinked to "base-1" in the quoted wikipedia page leads to radix which very clearly refers to "a system with radix $b$ ($\,\color{red}{b > 1}\,$)", therefore not applicable to the so-called "base-1" case.

Back to logarithms now, they do relate to the number of digits in the base-$b$ representation of numbers in positional notation with $b \gt 1$, because in that case each digit $\,d_k\,$ corresponds to a power $\,b^k\,$ of the base, and the rest follows from definitions. However, the unary representation is not a positional notation, but merely a bijective numeration one.

Therefore any expectation that the relation between logarithms and number of digits in positional notation with base $b \gt 1$ would somehow carry over to non-positional unary notation is misplaced.

dxiv
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Log is really about exponents, and the digit counting comes as a side effect. The reason log base $1$ is undefined is because the only positive number you can get by raising $1$ to some power is $1$. This is essentially why we can't divide by $0$ either.

The concept of base $1$ may function well in some applications similarly to other bases, but it breaks in this way because it's not actually a base, it's just a convenient, cool sounding name for writing a bunch of tally marks to represent a number. Those tally marks aren't digits.

Matt Samuel
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