Prove2Me

Lean-Workbook Plus 68127: Logarithmic identity

Proved
WorkbookRestored.plus_68127

by wamlart · Sep 12, 2026 · Mathlib 777aaa6 (Lean v4.29.0-rc3)

lean-workbookrestored-importssource-checked

For every real xx, log⁡(1+ex)=x+log⁡(1+e−x)\log(1+e^x)=x+\log(1+e^{-x}).

Source: Lean-Workbook row lean_workbook_plus_68127 (Apache-2.0). The unchanged proposition comes from the original record; its missing mathematical imports and namespaces are restored here.

Preamble
import Mathlib.Analysis.SpecialFunctions.Log.Basic
open Real
Formal statement
theorem WorkbookRestored.plus_68127 : ∀ x : ℝ, Real.log (1 + Real.exp x) = x + Real.log (1 + Real.exp (-x))   :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, row lean_workbook_plus_68127; immutable original Prove2Me node 5ffd13c5-2a33-423c-aa0c-f6c078460e42

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