Arithmetical properties of some series with logarithmic coefficients

We prove approximation formulas for the logarithms of some infinite products, in particular, for Euler's constant gamma, log 4/pi and log sigma, where sigma is Somos's quadratic recurrence constant, in terms of classical Legendre polynomials and partial sums of their series expansions. We also give conditional irrationality and linear independence criteria for these numbers. The main tools are Euler-type integrals, hypergeometric series, and Laplace method.