# Do the asymptotic frequencies of the first digits of $3\cdot 2^n$ and $2^n$ vary?

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.

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