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A given term of an expansion with logs or exponents equals a value: find x

This type of question, asked 3 times: 2021 Paper 7, 2021 Paper 17, 1 Feb 2023. All 3 questions below, oldest first.

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  1. Question 1

    2021 Paper 7

    70
    If the fourth term in the expansion of (x+xlog⁡2x)7\left( x+x^{\log_{2}x} \right)^{7} is 4480, then the value of xx where x∈Nx\in N is equal to:

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  2. Question 2

    2021 Paper 17

    62
    A possible value of ' xx ', for which the ninth term in the expansion of {3log⁡325x−1+7+3(−18)log⁡3(5x−1+1)}10\left\{ 3^{\log_{3}\sqrt{25^{x- 1}+ 7}}+3^{\left( -\frac{1}{8} \right)\log_{3}\left( 5^{x- 1}+ 1 \right)} \right\}^{10} in the increasing powers of 3(−18)log⁡3(5x−1+1)3^{\left( -\frac{1}{8} \right)\log_{3}\left( 5^{x- 1}+ 1 \right)} is equal to 180 , is:

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  3. Question 3

    1 Feb 2023

    174
    Let the sixth term in the binomial expansion of (log⁡2(10−33)+2xlog⁡235)m\left( \sqrt{\log_{2}\left( 10 -3^{3} \right)}+\sqrt[5]{2^{x\log_{2}3}} \right)^{m}, in the increasing powers of 2(x−2)log⁡232^{(x - 2)\log_{2}3}, be 21 . If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of A.P., then the sum of the squares of all possible values of xx is