Logarithm and Fractional Part

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 $k.$