Solve in $\mathbb{N}$ x $\mathbb{N}$ for $m$ and $n$ such that $m^n$ and $n^m$ have the same number of digits.

Question

Let $\delta : \mathbb{N} \rightarrow \mathbb{N}$ be a function such that $\delta (x)$ denotes the number of digits in $x$. Find all pairs of natural numbers $m$ and $n$ such that $\delta(m^n)=\delta(n^m)$.

I've tried using the formula $\delta(a^b) = \lceil a \log b \rceil $, but didn't get anywhere. More elegant methods / resources on this function would be greatly appreciated.