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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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    @ Alex & Jonas: for example by simple operations (such as scaling and shifting, eg. in the form of A*sin(B*log(C*x+D)+E).2011-04-08

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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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    I added a separate question for that: [link](http://math.stackexchange.com/questions/31712/sinx-suma-n-sinnlogxb-n-cosnlogx)2011-04-08