Number of Digits in Power of 2
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Theorem
Let $n$ be a positive integer.
Expressed in conventional decimal notation, the number of digits in the $n$th power of $2$:
- $2^n$
is equal to:
- $\ceiling {n \log_{10} 2}$
where $\ceiling x$ denotes the ceiling of $x$.
Proof
Let $2^n$ have $m$ digits when expressed in decimal notation.
By the Basis Representation Theorem and its implications, a positive integer $x$ has $m$ digits if and only if:
- $10^{m - 1} \le x < 10^m$
Thus:
\(\ds 10^{m - 1}\) | \(\le\) | \(\, \ds 2^n \, \) | \(\, \ds < \, \) | \(\ds 10^m\) | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds m - 1\) | \(\le\) | \(\, \ds \log_{10} \paren {2^n} \, \) | \(\, \ds < \, \) | \(\ds m\) | |||||||||
\(\ds \leadsto \ \ \) | \(\ds m - 1\) | \(\le\) | \(\, \ds n \log_{10} 2 \, \) | \(\, \ds < \, \) | \(\ds m\) |
Because a power of $2$ cannot equal a power of $10$, it will always be the case that:
- $m - 1 < n \log_{10} 2 < m$
and so:
- $m - 1 < n \log_{10} 2 \le m$
Hence from Integer equals Ceiling iff Number between Integer and One Less:
- $m = \ceiling {n \log_{10} 2}$
$\blacksquare$
Examples
Mersenne Number $M_{127}$
When expressed in conventional decimal notation, the number of digits in the Mersenne number $M_{127}$ is $39$.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $0 \cdotp 301 \, 029 \, 995 \, 663 \, 981 \ldots$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $0 \cdotp 30102 \, 99956 \, 63981 \ldots$