Logarithm Inequality

Algebra Level 5

Given that

\[ \begin{array} {lll} 0.161 & < \log_x 2 < & 0.162, \\ 0.256 & < \log_x 3 < & 0.257, \\ 0.375 & < \log_x 5 < & 0.376,\\ \end{array} \]

determine \( \lfloor \log_x 7^{100} \rfloor \).

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