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Algebra Level 5

\(x_{1} \times x_{2}... \times x_{n}=e\) and each \(\ln(x_{i})\) is a proper fraction. Find the greatest integral value of \[\ln(x)-\lfloor(\ln x_{1}) \times \ln(x)\rfloor -\lfloor(\ln x_{2}) \times \ln(x)\rfloor -\cdots-\lfloor (\ln x_{n}) \times \ln(x)\rfloor,\]

where \(n\) is the least 2-digit prime number, and \(\ln x \) is a natural number

Notation: \( \lfloor \cdot \rfloor \) denotes the floor function.

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