Given the set of all functions \begin{align} f_{a,b}(x): [0, \infty) &\to [0, \infty) \\ x &\mapsto a\log(1+bx) \end{align}
Is there a way of making the set of these functions $S =\{f_{a,b}: a,b \ge 0\}$ have the property that $\left[f_{a,b} + f_{a',b'}\right] \in S$? Can this be done by adding a fixed number of extra parameters to the definition?