Number of Binary Digits in Power of 10

Theorem
Let $n$ be a positive integer.

Expressed in binary notation, the number of digits in the $n$th power of $10$:
 * $10^n$

is equal to:
 * $\left\lceil{n \log_2 10}\right\rceil$

where $\left\lceil{x}\right\rceil$ denotes the ceiling of $x$.

Proof
Let $10^n$ have $m$ digits when expressed in binary notation.

By the Basis Representation Theorem and its implications, a positive integer $x$ has $m$ digits :
 * $2^{m - 1} \le x < 2^m$

Thus:

Because a power of $10$ cannot equal a power of $2$, it will always be the case that:
 * $m - 1 < n \log_2 10 < m$

and so:
 * $m - 1 < n \log_2 10 \le m$

Hence from Integer equals Ceiling iff Number between Integer and One Less:
 * $m = \left\lceil{n \log_2 10}\right\rceil$