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Value of c in Lagrange's mean value theorem for log x

The same question, asked 2 times: 2 May 2023 S2*, 10 May 2024 S2. Every wording below, oldest first.

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  1. Question 1

    2 May 2023 S2

    117
    The value of C for which the Mean Value Theorem holds for f(x)=log⁡exf(x) = \log_e x on [1,3][1,3] is

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  2. Question 2

    10 May 2024 S2

    131
    The value of cc for the function f(x)=log⁡xf(x)=\log x on [1,e][1,e] if LMVT can be applied, is

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