How to compute logarithms (without log function)

This article is a condensed version of the original, where the musical roots of the algorithm and its ancient origins are explored, making it less straightforward to follow. Here’s a clear explanation of how it works, think of it as a tutorial on “how to calculate logarithms in Python/JavaScript without the log function.”

The two Python implementations below are similar, except the second handles arguments in the 0–1 range. To make the logic explicit, both provide the list of consecutive convergents found, excluding the first one if integer.

def mla(a, b, max_q):

    if b <= 1 or a < 1:

        return []

    if a == 1:

        return ["0/1"]  # log_b(1) = 0

    if a == b:

        return ["1/1"]  # log_b(b) = 1

 for p in results]

    results = []

    link, lower, upper = a, 1, b

    p, q = 1, 1

    while q < max_q:

        while link < 1:

            link *= b; p += 1

        while link > b:

            link /= b; p += 1

        if q == 1:

            H = b / link; p += p - 1

        q += 1

        if link == b:

            results.append(f"{p-1}/1"); break

        if lower < link < upper:

            commas = [link/lower, upper/link]

            if (max(commas) - 1) / (min(commas) - 1) <= 2:

                results.append(f"{p}/{q}")

            lower, upper = (link, upper) if link < H else (lower, link)

        link *= a

    return results

The algorithm is a single dynamical template for different coordinate systems: additive (circle rotations), multiplicative (logs / exp / MLA),  angular (trig), and, implicitly, lattice / torus dynamics.

The core object is a rotation on a 1-torus, everything reduces to \(x \mapsto x+\alpha \pmod 1 \) for irrational \(\alpha\) : (Logs: \(\alpha = \log_b a\), angles: \(\alpha = \theta / 2\pi\), continued fractions: best rational returns of this rotation). The MLA is rotation dynamics written multiplicatively.

At a convergent \(p/q\): \(q\alpha \approx p\), the orbit nearly closes, the circle is partitioned into exactly two gap lengths. That’s the Three Gap Theorem at a convergent, this is why the ordering stabilizes, the gap structure simplifies, and discrete group actions suddenly appear. (The “dual cyclic groups”, at a convergent \(p/q\))

Object | Group | Generator

sorted indices | \(\mathbb{Z}/q\mathbb{Z}\) | \(p^{-1} \bmod q\)

overflow terms | \(\mathbb{Z}/p\mathbb{Z}\) | \(q^{-1} \bmod p\)

That symmetry is forced by \(p q' - q p' = \pm 1\) from CFs. So each convergent induces a pair of mutually dual cyclic actions, one “horizontal” (ordering) and one “vertical” (overflow). A lattice/tprus.

The “overflow” sequence is key, from the inequalities: \(\left|\alpha - \frac pq\right| < \frac{1}{q^2}\), MLA tracks modular advancement (floor terms, wrap counts) so that turns Diophantine approximation into explicit dynamics, group actions generated by irrational flow.

Trig case is the same object, replace multiplication with addition, \(\mathbb{R}^+/\langle b\rangle\) with \(S^1\) since \(a^q \approx b^p \quad \leftrightarrow \quad q\theta \approx 2\pi\).

All these computations (Pythagorean Scale/Sanfen-Sunyi, Grover Serach), are the same algorithm acting on different groups, with convergence limited by Diophantine structure.

TGT Details:

The three gap theorem says that for any irrational number \(\alpha\) and any positive integer \(n\), the set of fractional parts \(\{k\alpha\}\) (that is, \(k\alpha \bmod 1\)) for \(k = 1, 2, ..., n\), when arranged on the unit interval \([0,1)\), divides it into at most three distinct gap sizes. It also connects \(\alpha\) and the k values where gap changes relates to its continued fraction expansion. This algorithm uses a logarithmic isomorphism instead of \(k\alpha \bmod 1\); for instance, with \(\log_b(a)\), it looks at the modular remainders of exponential sequences \(a^x \cdot b^{y_x} \in (1, b]\), effectively rotating the \(a^x\) sequence within a modular space of \(b^{y_x}\). It detects when gaps change, marking that a convergent of the continued fraction expansion has been found. As the location of the change narrows, it focuses on comparing only the most recent gaps in that area.

This live demo calculates a few example convergents and renders one of the earlier ones. By default, it’s set to log2(3), showing a list of 10 convergents up to 1054/665, with the graphic displaying the fractional part of 19/12. (7/12)

For more on the music theory behind it, see (link), and for the theoretical basis (group theory, continued fractions, three gap theorem), see (link).