Transcendence of Logarithms
Summary: Euler’s §105 argument: if and are rational and is not a rational power of , then is neither rational nor irrational — i.e. not algebraic. Such quantities Euler calls transcendental. This is the chapter’s first extension of the algebraic/transcendental distinction from functions (Chapter 1) to numbers.
Sources: chapter6 (§105)
Last updated: 2026-04-26
The dichotomy
Suppose for some real , where are rational and is the base. Euler argues by cases on what kind of number could be (source: chapter6, §105):
- rational, . Then , i.e. . With both and rational, this forces to be exactly the -th power of — a rational power.
- irrational, . Then . Euler asserts this is impossible for rational — a power of a rational number with an irrational exponent cannot be rational.
So if is not a rational power of , then falls into neither category. By exclusion, is “neither rational nor irrational” in Euler’s sense — i.e. not algebraic. He coins the term transcendental for such quantities, and concludes:
Logarithms (in general) are transcendental.
What Euler is and is not proving
Euler’s argument is informal by modern standards. The second case — that cannot be rational for rational — requires the full Gelfond–Schneider theorem (1934) for a rigorous proof. Euler treats the impossibility as evident.
What Euler does establish cleanly is the dichotomy: either is a rational power of (in which case is rational) or is not rational. The “not rational” case he then calls transcendental, importing the language he had used for transcendental functions (see classification-of-functions) and applying it for the first time to a single quantity.
Why this matters
- It is the conceptual jump from transcendental functions (defined as not algebraic, in Chapter 1) to transcendental numbers (a number not satisfying any polynomial equation with rational coefficients).
- It justifies all the approximate methods that follow: §106 (geometric-mean iteration), §109 (tables of primes), §112 (decimal approximations via characteristic-and-mantissa).
- It explains why logarithm tables are useful: most logarithms cannot be written down exactly, so a table of decimal approximations is the only practical interface to them.
A working consequence
Almost every logarithm one encounters is transcendental — , , , , etc. — since these are only rational powers of 10 in the trivial cases (powers of 10 itself). Hence the entire table of common logarithms (Chapter 6, Briggs/Vlacq) consists of decimal approximations to transcendental numbers.