<p>In this article we investigate the property of complete monotonicity within a special family <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {F}_s\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">F</mi> <mi>s</mi> </msub> </math></EquationSource> </InlineEquation> of functions in <i>s</i> variables involving logarithms. The main result of this work provides a linear isomorphism between <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal {F}_s\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">F</mi> <mi>s</mi> </msub> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathcal {F}_s\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">F</mi> <mi>s</mi> </msub> </math></EquationSource> </InlineEquation> is semi-algebraic. This gives a finite time algorithm to decide whether a function in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathcal {F}_s\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">F</mi> <mi>s</mi> </msub> </math></EquationSource> </InlineEquation> is completely monotone.</p>