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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?

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No: the functions $\log(1+bx)$ for different $b > 0$ are linearly independent, so you would need infinitely many parameters. A way to see that they are linearly independent is to look at their derivatives, $\dfrac{b}{1+bx}$, which have poles at different points $x=-1/b$.

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Is it true that if the derivatives are linearly independent the functions are also linearly independet ? –  Theorem Jun 23 '12 at 7:12
    
Yes, because differentiation is linear: if a linear combination of the functions is $0$, the corresponding linear combination of the derivatives is also $0$. –  Robert Israel Jun 24 '12 at 6:16
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