Scaleless (or self-similar) function: $\sin ( \log x)$

Consider the function $f(x)=\sin( \log x)$ defined over $x>0$.

It has the cool feature that when you plot it, and change the x scale, it's overall shape does not change much. For example if you look at it over the $x$ range $[0,\;0.001]$ or $[0,\;1000]$ it's overall shape doesn't change.

Here is the question: Does there exist any positive real number $c$ that:

$f(c)=f(1)$ and $f(2c)=f(2)$ and $f(3c)=f(3)$ simultaneously?

Can we build a class of functions like $\sin ( \log x)$ that can form a base similar to fourier?

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Could you please elaborate on the meaning of the last question? What would meet the criteria of being "a class of functions such as $\sin(\log(x))$"? – Jonas Meyer Apr 8 '11 at 3:26
@svenkatr: I think he means a real number $c$ for which $\sin(\log(c))=\sin(\log(1))$, $\sin(\log(2c))=\sin(\log(2))$, and $\sin(\log(3c)) = \sin(\log(3))$. – Arturo Magidin Apr 8 '11 at 3:32
@Jonas: I think he means a set of such "similar" functions such that any periodic function can be approximated arbitrarily well by a linear combination of functions in the set. As for what counts as "similar", I imagine only the OP knows. – Alex Becker Apr 8 '11 at 3:33
@Alex: That is why I asked the OP :) @Arturo: $c=e^{2\pi}$ meets those criteria. As would $e^{2\pi\cdot n}$ for any integer $n$. – Jonas Meyer Apr 8 '11 at 3:33
@svenkatr: That comment should be an answer. – Rahul Apr 8 '11 at 3:38

2 Answers

Consider $c = e^{2 \pi}$. Then, $f(nc) = f(n)$ for all $n$.

EDIT: I had put this as a comment at first, but I made it an answer on Rahul Narain's suggestion.

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As for the second question, you could consider series in sin(n log x) and cos(n log x) for integers n, which correspond to Fourier series after the change of variable log x = t.

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Can you explain this a little more ? for example, can we expand $Sin(x)=\sum{a_n Sin(n log(x)+b_n Cos(n log(x))}$ ? – Austin Apr 8 '11 at 10:59
I added a separate question for that: link – Austin Apr 8 '11 at 11:40