# I want 1 to be first

Calculus Level 5

$$\displaystyle \{ a_n \}_{n=1}^{\infty}$$ be a sequence such that $$a_n = 2^{n-1}$$.

Define $$X(m) = \{ a_p | \text{ The first digit of } a_p \text{is } 1 ; 1 \leq p \leq m \}$$.

Let $$\displaystyle \{ b_n \}_{n=1}^{\infty}$$ be such that $$b_n$$ is the cardinality of $$X(n)$$.

The fraction $$\dfrac{b_n}{n}$$ is found to converge to $$L$$ as the value of $$n$$ becomes very large.

Evaluate: $$\displaystyle \lfloor 1000L \rfloor$$.

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