This a picture of the sequence A032740 from the OEIS (Numbers n such that n is a substring of 2^n.).
The graph is similar for A049301 (Numbers n such that n is a substring of 3^n.) and A049302 (Numbers n such that n is a substring of 4^n). For higher exponent bases, I think I see the same pattern but the data gets scarce.
In hindsight, it's not such a deep question.
The dip appears around when $a(n)=1000$. Suppose the number of digits in $a(n)$ is $m_n = \lfloor \log_{10}(a(n)) \rfloor$ and the number of digits in $2^{a(n)}$ is $k_n = \lfloor a(n) \log_{10}(2)\rfloor$.
The probability that a substring of $m_n$ integers appears in $k_n$ uniformly chosen random digits is simply
$$ \frac{(k_n - m_n + 1) 10^{k_n - m_n}}{ 10^{k_n}} = \frac{k_n - m_n + 1}{ 10^{m_n}} $$. Around when $a(n) = 1000$, $m_n$ suddenly increases by 1 and the probability decreases (less than) 10-fold. So the graph is steeper.
One can state more formally why each digit has nearly uniform distribution in the powers of 2.