  We introduce the class of \emph{Log-Noetherian} (LN) functions. These are
holomorphic solutions to algebraic differential equations (in several
variables) with logarithmic singularities. We prove an upper bound on the
number of solutions for systems of LN equations, resolving in particular
Khovanskii's conjecture for Noetherian functions. Consequently, we show that
the structure ${\mathbb R}_\text{LN}$ generated by LN-functions, as well as its
expansion ${\mathbb R}_\text{LN,exp}$, are effectively o-minimal: definable
sets in these structures admit effective bounds on their complexity in terms of
the complexity of the defining formulas.
  We show that ${\mathbb R}_\text{LN,exp}$ contains the horizontal sections of
regular flat connections with quasiunipotent monodromy over algebraic
varieties. It therefore contains the universal covers of Shimura varieties and
period maps of polarized variations of $\mathbb Z$-Hodge structures. We also
give an effective Pila-Wilkie theorem for ${\mathbb R}_\text{LN,exp}$-definable
sets. Thus ${\mathbb R}_\text{LN,exp}$ can be used as an effective variant of
${\mathbb R}_\text{an,exp}$ in the various applications of o-minimality to
arithmetic geometry and Hodge theory.
