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
$x$
.

The answer is 1506.

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Hint: $\lceil \log (5000^{5000})\rceil=18495$ just means that there are 18495 digits in the integer $5000^{5000}$ (in base 10).

Can we say a similar thing about $f(x)$ ?