Logarithms, constructible functions and integration on non-archimedean models of the theory of the real field with restricted analytic functions with value group of finite archimedean rank

Given a model of the theory real field with restricted analytic functions such that its value group has finite archimedean rank we show how one can extend logarithm to global values in polynomial ring over dimension rank. The logarithms are determined by algebraic data from model, namely section and an embedding into Hahn group. If coincides rational equivalent. We illustrate embed exponentiation. This allows us define constructible good lifting properties. As application establish Lebesgue measure integration ring, extending strengthening construction [T. Kaiser: on non-archimedean closed fields Proc. Lond. Math. Soc. 116 (2018), no. 2, 209-247.].