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Logarithms in A.P.: find the unknown

This type of question, asked 3 times: 2021 Paper 16, 24 Jan 2023, 29 Jan 2024. All 3 questions below, oldest first.

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  1. Question 1

    2021 Paper 16

    53
    If log⁡32,log⁡3(2x−5),log⁡3(2x−72)\log_{3}2,\log_{3}\left( 2^{x}- 5 \right),\log_{3}\left( 2^{x}-\frac{7}{2} \right) are in an arithmetic progression then the value of xx is equal to

  2. Question 2

    24 Jan 2023

    67
    For three positive integers p, q , r, xpq2=yqr=zp2rx^{pq^{2}}=y^{qr}=z^{p^{2}r} and r=pq+1r = pq + 1 such that 3,3log⁡yx,3log⁡zy3,3\log_{y}x,3\log_{z}y, 7log⁡xz7\log_{x}z are in A.P. with common difference 12\frac{1}{2}. Then r−p−qr - p - q is equal to

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  3. Question 3

    29 Jan 2024

    159
    If log⁡ea,log⁡eb,log⁡ec\log_{e}a,\log_{e}b,\log_{e}c are in an A.P. and log⁡ea−\log_{e}a - log⁡e2b,log⁡e2b−log⁡e3c,log⁡e3c−log⁡e\log_{e}2b,\log_{e}2b -\log_{e}3c,\log_{e}3c -\log_{e} a are also in an A.P, then a:b:ca:b:c is equal to

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