Foundation Logic Real Transcendentals Log L Exp L

The inverse pair

The natural logarithm and the exponential function are inverses: for any positive number x, e^{ln x} = x, and for any real number x, ln(e^x) = x. This pairing is the bridge between multiplication and addition that makes logarithms useful for compressing scales, solving growth problems, and turning products into sums. The second identity, ln(e^x) = x, is the one at issue here. It states that applying the exponential first and the logarithm second returns you to your starting point, no matter which real number you began with.

This fact has a long history. John Napier published his description of logarithms in 1614, and the modern exponential notation arrived with Leonhard Euler in the eighteenth century. The inverse relationship is what lets a slide rule multiply by adding lengths, and it underpins the continuous growth models of finance and physics. The identity holds for the ordinary real numbers, and it is a standard result of analysis.

In Recognition Science, the framework's machine-checked library of formal results reconstructs the real-number line as its own object, called LogicReal, and shows it is equivalent to the usual real numbers. The declaration logL_expL is the formal statement that, on this reconstructed line, the logarithm of the exponential of any number returns that number exactly. The derivation is short: it rewrites the transported definitions and then applies the standard result from the underlying real-analysis library. The same library also shows the companion identity, that exponentiating the logarithm of a positive number returns the number itself.

What this declaration does not claim is just as important. It does not show that the logarithm and exponential are inverse functions in any new or independent sense; the content is entirely inherited from the standard theory. It does not establish the existence or uniqueness of these functions from the framework's axioms. It is a transport lemma, a piece of bookkeeping that lets later modules reason about the reconstructed line using the familiar analytic facts. The value is practical: it confirms the framework's real numbers behave like the real numbers, so that subsequent work can use logarithms and exponentials without re-deriving their properties.

/-- Transported logarithm/exponential inverse. -/
theorem logL_expL (x : LogicReal) : logL (expL x) = x := by
  rw [eq_iff_toReal_eq, toReal_logL, toReal_expL]
  exact Real.log_exp _

/-- Transported logarithm/exponential inverse. -/
theorem logL_expL (x : LogicReal) : logL (expL x) = x := by
  rw [eq_iff_toReal_eq, toReal_logL, toReal_expL]
  exact Real.log_exp _

/-- Transported logarithm/exponential inverse. -/
theorem logL_expL (x : LogicReal) : logL (expL x) = x := by
  rw [eq_iff_toReal_eq, toReal_logL, toReal_expL]
  exact Real.log_exp _

What this page does not claim

It does not show the inverse property in any sense independent of the standard real-analysis library. It does not derive the existence of the exponential or logarithm from the framework's axioms. It does not establish any new property of the ordinary real numbers.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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