On the $b$-ary expansions of $\log (1 + \frac{1}{a})$ and ${\mathrm e}$

Let $b \ge 2$ be an integer and $ξ$ an irrational real number. We prove that, if the irrationality exponent of $ξ$ is equal to $2$ or slightly greater than $2$, then the $b$-ary expansion of $ξ$ cannot be `too simple', in a suitable sense. Our result applies, among other classical numbers, to badly approximable numbers, non-zero rational powers of ${\mathrm e}$, and $\log (1 + \frac{1}{a})$, provided that the integer $a$ is sufficiently large. It establishes an unexpected connection between the irrationality exponent of a real number and its $b$-ary expansion.

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