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:

\(\displaystyle 10^{m - 1}\) \(\le\) \(\, \displaystyle 2^n \, \) \(\, \displaystyle <\, \) \(\displaystyle 10^m\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle m - 1\) \(\le\) \(\, \displaystyle \log_{10} \paren {2^n} \, \) \(\, \displaystyle <\, \) \(\displaystyle m\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle m - 1\) \(\le\) \(\, \displaystyle n \log_{10} 2 \, \) \(\, \displaystyle <\, \) \(\displaystyle m\)

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