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Math Help - Musings concerning maximal expressions leading to a logarithmic equation

  1. #1
    Junior Member NowIsForever's Avatar
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    Lightbulb Musings concerning maximal expressions leading to a logarithmic equation

    Upon awakening this morning I had this idea about the largest expression that could be formed from a fixed collection of digits, all the same number, using standard operations such as multiplication, exponentiation, etc.

    For example suppose, the collection is 2,2,2,2. My first thought was that 2^2^2^2 would be the answer for the maximum value, however 22^2^2 is larger, and 2^2^2^2 is larger still.

    Then I noticed that 22^2^2 is closer to 2^2^2^2 (by ratio) than 33^3^3 is to 3^3^3^3. The situation is further divergent as the value of the digit climbs to 9.

    Define d*d as 10d+d and d*d*d as d100+d10+d etc., where d is not only allowed to be a digit, but any real number. (Numbers less than or equal to zero, although allowed, may or may not be reasonable).

    So this thought formed: what would be the value of d such that (d*d)^(^d^*^d^) = d^(^d^*^d^*^d^)?

    It would be a solution to the equation: (10x+x) log_x (10x+x)= 100x+10x+x.

    Not wishing to look for an analytic solution to the equation prior to determining a numerical result; I proceeded to do that obtaining

    x = 1.3018267624863938102032556203448

    using my Windows calculator after 39 iterations (I know, I could have written a C program to do this in less than half the time, but I'm on vacation for three weeks, I like doing it, and I probably would not have discovered the following result had I written the program):

    I noticed this as a convergent during my calculations: log_x (10x+x) = 111/11.

    It is not obvious to me why this should be so; perhaps it would fall out of an analytic solution quite easily, but I have yet to look for one, and I'm posting this so that all and sundry may have a shot at it as I work on it myself.

    Any feedback would be greatly appreciated.
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  2. #2
    Junior Member ignite's Avatar
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    Re: Musings concerning maximal expressions leading to a logarithmic equation

    x=11^\frac{11}{100} solves the given equation.
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