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In my class we have shown that the asymptotic frequency of $d$ being the first digit(s) of $2^n$ is $f(d)=log_{10}(\frac{d+1}{d})$.

What happens to the asymptotic frequency when instead we consider $3\cdot2^n$?

I'm pretty sure the frequencies should remain unchanged, as exchanging the quantity $2^n$ to $3\cdot 2^n$ doesn't appear to alter the argument we made in class.

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    $\begingroup$ Could you reproduce at least the outline of the argument you had in class? We can't really answer this question without that context. $\endgroup$ – Milo Brandt Feb 20 at 4:02

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