Everything Is Logarithms: A Unified Theory of Multiplicative-to-Additive Isomorphisms
The Baseless Logarithm: Logarithms as Coordinate-Free Objects
Logarithms are fundamentally isomorphisms that translate multiplicative relationships into additive ones. While standard notation $\log_b(x)$ ties a logarithm to a specific base, the concept of a "baseless logarithm" treats the logarithm as an abstract geometric object—a point—rather than a specific number. In this framework, a based logarithm is simply a ratio of two baseless logarithms:
$$\log_b N = \frac{\log N}{\log b}$$
This perspective transforms the change-of-base formula from an algebraic rule into a simple change of units. Just as 2 kilometers is 2000 meters, $\log_2 N$ (measured in "bits") can be rewritten as $\ln N$ (measured in "nats") by dividing by the unit $\log e$. This mirrors the distinction in vector calculus between a displacement vector (the difference between two points) and a point itself; the baseless logarithm is the fundamental point, and the based logarithm is the displacement once a coordinate system (the base) is chosen.
Logarithms as Vectors and Projections
There is a direct structural equivalence between baseless logarithms and geometric vectors. A geometric vector $\mathbf{v}$ exists independently of any coordinate system, but can be expressed as a collection of numbers once projected onto a basis vector $\mathbf{x}$ (e.g., $v_x = \mathbf{v}/\mathbf{x}$). Similarly, the baseless logarithm $\log N$ is a geometric object that becomes a numeric value when projected onto a "measuring stick" like $\log 2$.
Logarithmic Projections in Mathematics
While standard logarithms provide a total value, several mathematical fields have independently developed "projection" operators that extract specific logarithmic components, mimicking partial derivatives ($\partial f / \partial x$):
- Number Theory: The $p$-adic valuation $\nu_p(n)$ extracts the exponent of a prime $p$ in the prime factorization of $n$. This is effectively a projection of the "vector" $\log n$ onto the basis element $\log p$.
- Complex Analysis: The "order of vanishing" $\text{ord}a f(z)$ of a meromorphic function at point $a$ is extracted using the limit $\lim{z \to a} \frac{\log f(z)}{\log(z-a)}$, which serves to cancel out other terms and isolate a specific component, much like a partial derivative.
Vectors as Logarithms of Translation Operators
In differential geometry, vectors are often written as a basis of partial derivative operators: $\mathbf{v} = v_x \partial_x + v_y \partial_y$. These operators generate discrete translations via exponentiation:
$$T_{\mathbf{v}} = e^{v_x \partial_x + v_y \partial_y} = T_x^{v_x} T_y^{v_y}$$
This reveals that vectors are, in essence, the logarithms of translation operators. By treating the translation operator $T$ as a generic base, a vector $\mathbf{v}$ can be viewed as $\log_T T_{\mathbf{v}}$. This mapping converts the multiplicative action of translating a point in space into the additive algebra of vector components.
The Logarithmic Nature of Dimension
The dimension operator ($\text{dim}$) in linear algebra behaves exactly like a logarithm. This is evident when comparing the properties of vector space operations to logarithmic identities:
| Vector Space Operation | Dimension Result | Logarithmic Equivalent | Log Result |
|---|---|---|---|
| Direct Sum $U \oplus V$ | $\text{dim}(U) + \text{dim}(V)$ | Multiplication $u \times v$ | $\log(u) + \log(v)$ |
| Quotient Space $U/V$ | $\text{dim}(U) - \text{dim}(V)$ | Division $u/v$ | $\log(u) - \log(v)$ |
| Tensor Product $U \otimes V$ | $\text{dim}(U) \times \text{dim}(V)$ | Exponentiation $u^{\log v}$ | $\log(u) \times \log(v)$ |
For finite-dimensional vector spaces over a finite field $K$, this is literally true: the cardinality of the vector space $V$ is $|K|^{\text{dim}_K V}$. Therefore, $\text{dim}K V = \log{|K|} |V|$. This suggests that the $\text{dim}$ and $\text{span}$ operators are the linear algebra analogues of $\log$ and $\exp$.
Synthesis and Theoretical Implications
These parallels suggest a unifying theory where many disparate mathematical operations are actually instances of the same primitive: the isomorphism between multiplicative and additive representations.
Insights from the Community
Discussion around these concepts highlights several theoretical connections:
- Torsors: Commenters noted that the "baseless log" is a torsor—an algebraic object that is like a group but lacks a fixed identity element (origin). Choosing a base for a logarithm is equivalent to "trivializing" the torsor by picking an origin.
- Physics: In quantum mechanics and statistical mechanics, the action $S$ and entropy $S$ both function as logarithms that convert multiplicative probabilities or amplitudes into additive quantities.
- Critique: Some argue that while these connections are aesthetically pleasing, they may be "macro-level overgeneralizations" that risk flattening the practical differences between tools designed for different problems.
Ultimately, the "baseless logarithm" approach advocates for general covariance in mathematics—the idea that the fundamental properties of mathematical objects should be independent of the coordinates (or bases) used to describe them.