Logarithms as Vectors: Rethinking Bases, Bits, and Nats

A math blog post proposes treating the logarithm as a coordinate-free object. Instead of writing log_b(x) with a fixed base, the author introduces a “baseless logarithm” log N as an abstract quantity, then recovers an ordinary logarithm as the ratio of two of them: log_2(N) = log(N)/log(2). Under this view, log(2) becomes a unit — “bits” — and log(e) becomes “nats,” so the familiar change-of-base formula is just re-expressing the same quantity in different units, much like converting kilometers to meters or bytes to bits.

The core claim is that logarithms behave like vectors. A geometric vector has meaning only once you project it onto a basis or “measuring stick,” and its numeric components are that projection. The baseless logarithm plays the same role multiplicatively: log N is the abstract “point,” the chosen base is the measuring stick, and dividing one by the other cancels an unspecified origin to yield a concrete number. The author draws the analogy out to displacement vectors (defined relative to an arbitrary origin) and to partial derivatives as a kind of pseudo-division that extracts a component.

The piece is exploratory rather than rigorous — the author openly frames these as noticed connections that “maybe mean something.” One conclusion worth flagging: there is no useful “baseless exponential,” because splitting log_b(N) into separate log N and log b factors leaves each piece unitless and without standalone numeric value. The appeal is pedagogical, offering a cleaner mental model for why logarithms and unit conversions feel structurally alike.