# Log First

Algebra Level 5

If $$\ln x_1+ \ln x_2 + \cdots + \ln x_n =1$$, where 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 $$n = 101$$, and $$\ln x$$ is an integer.

Notation: $$\lfloor \cdot \rfloor$$ denotes the floor function.

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