We have all natural numbers whose decimal expansion has only even digits $0,2,4,6,8$.
$a_n=n$th number of sequence if these are arranged in increasing orders.
Let $n=5^K$, then $5$ based representation of $n$ is a $1$ followed by $K$ zeroes,
$\therefore$ Decimal representation and doubling its value, we obtain a $2$ followed by $K$ zeroes that is $210^K$.
$\therefore \quad \log a_n=K+\log _{10} 2$
$\Rightarrow \log n =K \log _{10} 5$
$\Rightarrow \quad \lim _{n \rightarrow \infty} \frac{\log a_n}{\log n} =\frac{K+\log _{10} 2}{K \log _{10} 5}$
$K$ goes to infinity
$\therefore \quad \lim _{n \rightarrow \infty} \frac{\log a_n}{\log n}=\frac{1}{\log _{10} 5}=\log _5 10$
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