Chance of the first digit of $2^{n}$ being $1$?

Asked Jan 28 '19 at 05:58

Active Jan 28 '19 at 05:58

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What's the probability that the first digit of $2^{n}$ is $1$, where $n$ is uniformly distributed over all integers?

I've been running simulations in MATLAB, and I think the answer converges to $\log(2)$, but I have no clue how to prove this. Any ideas?

asked Jan 28 '19 at 05:58

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"Uniformly distributed over all integers" is not a well defined probability assignment. Instead, consider being uniformly defined over $[1,N]$ for some large number $N$ and taking $N \to \infty$. – Sarvesh Ravichandran Iyer Jan 28 '19 at 06:03
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Also see here : https://math.stackexchange.com/questions/417486/probability-that-the-first-digit-of-2n-is-1 You may want to read up Benford's law as well. – Sarvesh Ravichandran Iyer Jan 28 '19 at 06:13
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In binary it is $1$. – Ross Millikan Jan 28 '19 at 06:13

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