Does $\log_1{n}$ have any analytical significance in counting digits of unary numbers?

Question

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?

Answer 1

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.

Answer 2

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.