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Sum of log x to the base a^(1/k) over many terms

This type of question, asked 2 times: 2021 Paper 6, 2021 Paper 12. All 2 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2021 Paper 6

    90
    Let Sn(x)=log⁡a1/2x+log⁡a1/3x+log⁡a1/6xS_{n}(x) =\log_{a^{1/2}}x+\log_{a^{1/3}}x+\log_{a^{1/6}}x
    +log⁡a1/11x+log⁡a1/18x+log⁡a1/27x+…+\log_{a^{1/11}}x+\log_{a^{1/18}}x+\log_{a^{1/27}}x+ \ldots
    up to n-terms, where a>1a> 1. If S24(x)=1093S_{24}(x) = 1093 and S12(2x)=265S_{12}(2x) = 265, then value of aa is equal to

  2. Question 2

    2021 Paper 12

    57
    If sum of the first 21 terms of the series log⁡91/2X+log⁡91/3X+log⁡91/4X+…\log_{9^{1/2}}X +\log_{9^{1/3}}X +\log_{9^{1/4}}X + \ldots. , where x>0x > 0 is 504 , then xx is equal to

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