logarithmic singularities in contour integration

Question

How to evaluate the contour integral using the residue theorem if there is a logarithmic derivative? For example this: $$\int_C \log\zeta(s)\frac{x^s}{s} ds$$ or even this: $$ \int_C \frac{\log x}{x}dx$$ (I'm not interested in how to evaluate them, but in how the $\log$ affects them and/or how to evaluate integrals with logarithms)