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

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