Classifying spaces of degenerating mixed Hodge structures, VI: Log real analytic functions and log $C^{\mathrm{\infty }}$ functions

To advance our log Hodge theory, we introduce log real analytic functions and log $C^{\mathrm{\infty }}$ functions, define how to integrate them, and prove the log Poincaré lemma. We give better understandings of the degeneration of Hodge structure, including a geometric interpretation of the theory of $SL(2)$-orbits.