# 5000 Floors, Ceilings, Fracs, Logs, Powers

Let $f(x)=\left\lfloor10^{\{x\log (5000)\}}\right\rfloor$

There exists $$n$$ distinct non-negative integer values $$k_1, k_2, \ldots k_n < 5000$$ such that $$f(k_i)=1$$.

Given that $$\lceil \log (5000^{5000})\rceil=18495$$, find $$n$$.

Details and Assumptions

$$\lfloor x \rfloor$$ is the largest integer smaller than $$x$$.
$$\lceil x \rceil$$ is the smallest integer larger than $$x$$.
$$\{x \}=x-\lfloor x \rfloor$$ is the fractional part of $$x$$.

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