Logarithm and Fractional Part

Algebra Level 4

A sequence \(\{a_n\}\) whose elements are all positive satisfies the following relationship:\[a_1\times a_2^2\times a_3^3\times \cdots \times a_n^n=10^{n^2-n} \text{ for all positive integers } n.\] When the fractional part of \(\log a_k\) is \(0.99,\) find the value of \(k.\)

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