Everything is Logarithms: Exploring Baseless Logarithms and Mathematical Covariance

The Core Thesis: Logarithms as Coordinate-Free Objects

Logarithms are fundamentally isomorphisms that translate multiplicative algebraic representations into additive ones. By treating the logarithm as a "baseless" geometric object rather than a specific numerical value, we can view the choice of a base (e.g., base 2 for bits, base $e$ for nats) as a choice of units or a coordinate system. This perspective reveals that many disparate mathematical operations—from the dimension of a vector space to $p$-adic valuations—are actually instances of the same underlying logarithmic primitive.

The Baseless Logarithm and Change of Units

In standard notation, $\log_b(x)$ is a number. However, if we treat $\log N$ as an abstract, baseless object, the standard "based" logarithm becomes a ratio of two baseless logarithms:

$$\log_2 N = \frac{\log N}{\log 2}$$

In this framework, $\log 2$ is interpreted as the unit "bits." Changing the base of a logarithm is therefore not just an algebraic manipulation but a change of units, analogous to converting kilometers to meters. This mirrors the distinction in vector calculus between a geometric vector (an abstract displacement) and a coordinate vector (a tuple of numbers relative to an origin).

Logarithms as Vectors and Projections

There is a strong structural equivalence between baseless logarithms and geometric vectors. Just as a vector $v$ can be projected onto a basis vector $x$ to find its coordinate $v_x$, a baseless logarithm $\log N$ can be projected onto a unit $\log 2$ to find the value $\log_2 N$.

Logarithmic Projections in Other Fields

While standard logarithms lack a direct "partial derivative" operator, other mathematical fields have independently invented logarithmic projections:

• Number Theory: The $p$-adic valuation $\nu_p(n)$ extracts the coefficient of $\log p$ in the logarithmic basis of a natural number, effectively acting as a projection.
• Complex Analysis: The "order of vanishing" $\text{ord}a f(z)$ of a meromorphic function is extracted using a limit of the ratio of logarithms: $\lim{z \to a} \frac{\log f(z)}{\log(z-a)}$.

Vectors as Logarithms of Translation

In differential geometry, vectors are often written as partial derivative operators. A translation operator $T_v$ can be expressed as the exponential of a vector: $T_v = e^v$.

Conversely, a vector can be viewed as the logarithm of a translation operator: $v = \ln T_v$. By using the "baseless" approach, we can write $v = \frac{\log T_v}{\log T}$, where $T$ is a generic base for translations. This suggests that the very concept of a vector in flat space is the logarithm of a multiplicative translation operation.

Dimension as a Logarithm

The dimension operator $\text{dim}_K V$ in linear algebra behaves exactly like a logarithm. The following correspondences hold:

• Direct Sum $\to$ Addition: $\text{dim}_K(U \oplus V) = \text{dim}_K U + \text{dim}_K V$
• Tensor Product $ o$ Multiplication: $ ext{dim}_K(U \otimes V) = \text{dim}_K U \times \text{dim}_K V$

This is not merely an analogy. For finite-dimensional vector spaces over finite fields, the dimension is literally the logarithm of the logarithm of the cardinality of the space relative to the logarithm of the cardinality of field: $\text{dim}K V = \log{|K|} |V|$.

Synthesis and Discussion

This "setification" of arithmetic—treating numbers as cardinalities of sets and operations as set-theoretic constructions—suggests that much of mathematical notation is obscuring a deeper, more unified structure.

Community Perspectives

Discussion among technical peers highlights both the elegance and the risks of mathematical generalization:

"The baseless log here here is just a torsor! ... Torsors let us talk about these things without needing to make an arbitrary choice a priori."

While some argue that this is a powerful way to kilometers to meters.

"Saying everything is just one big logarithm is a nice mental exercise, but I feel like it flattens out the differences too much and makes you lose the practical utility of the individual math tools."

Ultimately, the bit-level of mathematics is a covariant formulation where properties are properties of relations between measurements rather than absolute values, reducing the redundancy in disparate notations.