5000 Floors, Ceilings, Fracs, Logs, Powers

Number Theory Level 5

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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