Foundation Logic Real Transcendentals Exp L Log L

The exp-log inverse pair

The exponential function and the natural logarithm undo each other. For any positive number x, raising e to the power of the natural log of x returns x itself: e^ln(x) = x. This is the defining relationship of the logarithm, which asks the question "to what power must e be raised to produce this number?" The exponential answers that question directly. Their inverse relationship is what makes logarithms useful for turning multiplication into addition and for solving equations where the unknown sits in an exponent.

This pair of functions has a long history. John Napier published the first tables of logarithms in 1614, and the natural logarithm with base e was recognized in the century that followed. The constant e itself, approximately 2.71828, emerged from the study of continuously compounded interest. The functions appear throughout science: exponential growth describes populations and chain reactions, exponential decay describes radioactive substances and capacitor discharge, and the logarithm compresses wide-ranging values into human-scale numbers, as on the decibel scale for sound or the Richter scale for earthquakes.

In Recognition Science, the framework constructs its own copy of the real number line, called the recovered reals, and then transports the standard real-analysis functions onto it. The declaration expL_logL is a theorem in the framework's machine-checked library of formal theorems. It states that on the recovered reals, the transported exponential function applied to the transported natural logarithm of a positive number returns that same number. The proof is a direct transfer: it converts the statement to the familiar real numbers, where the corresponding fact is already established, and applies that known result.

The framework's library also proves the reverse direction: the natural logarithm of the exponential of any number returns that number. Together these two theorems establish that the transported functions form a genuine inverse pair on the recovered reals, not merely a pair of functions that happen to share names with their classical counterparts. This matters because later modules in the library reason over the recovered reals while relying on the analytic identities of the standard real numbers. The inverse pair is a small but load-bearing piece of that bridge.

What the declaration does not claim is equally precise. It does not assert that the exponential or logarithm functions are new or different on the recovered reals; they are defined by transport from the standard real numbers, so they inherit all their properties. It does not claim that the inverse relationship holds for non-positive inputs, since the logarithm is only defined for positive numbers. And it makes no statement about the philosophical status of the recovered reals themselves; that is a separate matter in the framework's development.

/-- Transported exponential/log inverse on positive recovered reals. -/
theorem expL_logL {x : LogicReal} (hx : (0 : LogicReal) < x) :
    expL (logL x) = x := by
  rw [eq_iff_toReal_eq, toReal_expL, toReal_logL]
  have hx' : 0 < toReal x := by simpa [lt_iff_toReal_lt] using hx
  exact Real.exp_log hx'

/-- Transported exponential/log inverse on positive recovered reals. -/
theorem expL_logL {x : LogicReal} (hx : (0 : LogicReal) < x) :
    expL (logL x) = x := by
  rw [eq_iff_toReal_eq, toReal_expL, toReal_logL]
  have hx' : 0 < toReal x := by simpa [lt_iff_toReal_lt] using hx
  exact Real.exp_log hx'

/-- Transported exponential/log inverse on positive recovered reals. -/
theorem expL_logL {x : LogicReal} (hx : (0 : LogicReal) < x) :
    expL (logL x) = x := by
  rw [eq_iff_toReal_eq, toReal_expL, toReal_logL]
  have hx' : 0 < toReal x := by simpa [lt_iff_toReal_lt] using hx
  exact Real.exp_log hx'

What this page does not claim

The declaration does not define new transcendental functions; they are transported from the standard real numbers. The inverse relationship does not hold for non-positive inputs, since the logarithm is undefined there. The theorem makes no philosophical claim about the recovered reals themselves.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/LogicRealTranscendentals.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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