# 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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