Friend Exponent

For given two real numbers $\alpha$ and $\beta$, a positive integer $n$ is called $k$-friend exponent of $\alpha$ for $\beta$ if ${\alpha }^n$ and $\beta$ shares the first $k$-significant digits.

For example, $n=9$ is a $1$-friend exponent of $\pi$ for $e$ because ${\pi }^9=29809.099333446...$ and $e=2.71828...$ shares "$2$".

Let $E(\alpha ,\beta ,k)$ denote the minimum $k$-friend exponent of $\alpha$ for $\beta$. It can be verified that $E(\pi ,e,1)=9$, $E(\pi ,e,2)=19$, $E(\pi ,e,5)=430881$ and $E(\pi ,e,10)=2448322414$.

Find $E(\pi ,e,50)$.

[My timing: 0.5 sec. (PyPy)]