Abstract In this article we investigate the property of complete monotonicity within a special family $$\mathcal {F}_s$$ F s of functions in s variables involving logarithms. The main result of this work provides a linear isomorphism between $$\mathcal {F}_s$$ F s and the space of real multivariate polynomials. This isomorphism identifies the cone of completely monotone functions with the cone of non-negative polynomials. We conclude that the cone of completely monotone functions in $$\mathcal {F}_s$$ F s is semi-algebraic. This gives a finite time algorithm to decide whether a function in $$\mathcal {F}_s$$ F s is completely monotone.