Number of Digits in Power of 2

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 :
 * $10^{m - 1} \le x < 10^m$

Thus:

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:
 * $m = \ceiling {n \log_{10} 2}$