Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
up vote 2 down vote accepted

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 $x=-1/b$.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.