Number of Digits to Represent Integer in Given Number Base

Theorem
Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $b \in \Z$ be an integer such that $b > 1$.

Let $d$ denote the number of digits of $n$ when represented in base $b$.

Then:
 * $d = \left\lceil{\log_b \left({n + 1}\right)}\right\rceil$

where $\left\lceil{\,. \,}\right\rceil$ denotes the ceiling function.

Proof
Let $n$ have $d$ digits.

Then: