Logarithm with variable base

Question

I am trying to define a function that maps polynomials in the form of $x^{3^n}$ to the value of $n$ in the polynomial, where $n\in{Z}$.* Is is valid to define this function as $log_{x^3}(u)$, where $u \in [{x^{3^n}} : n\in{Z}]$?

On the one hand I'm uncomfortable because I'm putting a variable in the base of a logarithm, but on the other hand, I don't think it matters because that variable doesn't depend on the input or output of the logarithm, so we can think of it like a constant base. Does it make sense to define the function this way?

*If anyone could show me how to write a blackboard Z in latex, that would be great.

Answer 1

It's perfectly acceptable to have variable bases for logarithms. Keep in mind that $\log_ba=\frac{\log a}{\log b}$. Therefore, $\log_{x^3}u=\frac{\log u}{\log x^3}$, so you don't even have to think of it as being a variable base.

Answer 2

The base $x$ logarithm $log_x$ takes a function $f(x)$ in $x$ and produces a numeric result in such a way that $log_x x^n = n$ for all values of $n$, including non-integer values. So, one possible answer is: $$log_x f(x) = \lim_{h,k→0} {1 \over k}\left({{f(h(1+k)) \over f(h)} - 1}\right).$$

To handle $log_{x^3}$, we can generalize to $log_{g(x)}(f(x))$, where $g(x)$ is one-to-one, by defining: $$log_{g(x)}(f(x)) = log_x(f(g^{-1}(x))),$$ so that: $$log_{x^3}(f(x)) = log_x(f(\sqrt[3]{x})).$$

It can be further generalized to $nOf(f,a) = log_x(f(x-a))$, which either (a) yields a positive value that counts the multiplicity of the root $x = a$, if $f(a) = 0$ (including fractional multiplicities, like for $\sqrt{x-a}$), (b) yields a negative value that indicates the negative of the degree of singularity of $f(x)$ at $x = a$, if $f(x) → ∞$ as $x → a$ or (c) yields 0, if neither condition applies. Then $nOf(f,0) = log_x f(x)$ and $nOf(f,0) = n$ when $f(x) = x^n$.

It might also be possible to prove that this answer is unique, subject to generic, natural conditions; e.g. $log_x(f(x)g(x)) = log_x(f(x)) + log_x(g(x))$, $log_x 1 = 0$, $log_x x = 1$, and maybe other similar conditions.

One example - which prompted my query into the matter - is $nOf(j_n) = n$ for the Bessel functions $j_n(x)$, including those of half-integer order $n$. The idea of defining these functionals came in a dream, early this morning, out of the blue. I've never seen anything like this before, so I have no reference to any of this other than to myself.