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Math Help - Maximum value of x in an equation: Involves constant to the power of x

  1. #1
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    Maximum value of x in an equation: Involves constant to the power of x

    Hi everyone, this is part of a longer question, but the current part involves me finding the value of c in the following equation:
    L = c^3.(0.5)^(c-1).(0.6)^(c-1).(0.8)^(c-1).(1- (0.5)^c)^42.(0.5)^(8c)


    Here's what I've done so far:

    I take the natural log of both sides:

    ln L = 3.ln(c) + (c-1)ln(0.5) + (c-1)ln(0.6) + (c-1)ln(0.8) + 42ln(0.5^c - 1) + 8c.ln(0.5)

    Then differentiate to find the max:

    ∂(lnL)/∂c = 3/c + ln(0.5) + ln(0.6) + ln(0.8) + [42[ln(0.5)](0.5)^c]/(0.5^c - 1) + 8ln(0.5)

    At max, ∂(lnL)/∂c = 0

    Relabel [ln(0.5) + ln(0.6) + ln(0.8) + 8ln(0.5)] as (-A) for convenience

    Therefore,

    3/c + [42[ln(0.5)](0.5)^c]/(0.5^c - 1) = A

    Multiplying across by c(0.5^c - 1) gives:
    3
    (0.5^c) - 3 + 42.ln(0.5)c(0.5)^c = Ac(0.5)^c - Ac

    Relabel 42.ln(0.5) as B for convenience

    Rearranging, we get:

    (3 + Bc - Ac)(0.5)^c = 3 - Ac

    c.ln(0.5) = ln ((3-Ac)/(3 + Bc + Ac))

    ---

    And this is where I run into problems, as I can't get rid of the ln on the right side without making an e^c on the left, so I don't know how to isolate the c. I tried differentiating both sides, but I'm pretty sure you can't do that. Is my entire method wrong? The equation at the start is definitely correct though. Please could someone point me in the right direction?
    Thanks!
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  2. #2
    MHF Contributor

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    That cannot be solved in terms of "elementary functions" though you might be able to solve it using "Lambert's W function" which is defined as the inverse function to f(x)= xe^x.
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