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Can someone just help explain a clear defination of superlog?

  1. Mar 27, 2012 #1
    I was trying to understand superlog and superroot but I get only 3/4 of them. Can anyone just explain, in a non-textbook way, such that:
    I can understand without any post-calc knowledge
    http://en.wikipedia.org/wiki/Superlog
    -------------------------------------------------
    or just explain:
    c=x^x
    how to find x?
    or c=x^x^x^x^x^x^x.............
    thanks. just to say again that im not even doing calc in school, but Im doing all maths myself. So will you explain without using any complex analysis and other things?
     
  2. jcsd
  3. Mar 27, 2012 #2

    chiro

    User Avatar
    Science Advisor

    Hey n_kelthuzad and welcome to the forums.

    The reason why we need complex analysis is because the solution x could actually be negative. When x is negative then we are looking at most likely a complex number solution depending on what x actually is.

    So lets for the moment assume that x > 0. We only consider a finite number of exponentiations: Let's look at two first.

    c = x^x^x. Assume x > 0 and c > 0 (It has to be otherwise it won't work).

    log_c(c) = log_c(x^x^x) = 1

    But log_c(x^x^x) = x^xlog_c(x) = 1.

    This meanx x^x = 1/log_c(x).

    Now lets do the same thing again:

    log_c(x^x) = log_c(1/log_c(x)) = - log_c(log_c(x)) since log(1) = 0 and log(a/b) = log(a) - log(b).

    But log_c(x^x) = xlog_c(x) which means:

    xlog_c(x) = -log_c(log_c(x)) which means

    x = -log_c(log_c(x)) / log_c(x)

    To solve this, you need to use a computer for the general case. Now log_c(x) has to be greater than 0 otherwise you won't get a solution. This means that x > c for the real case (remember no complex numbers).

    Since log_c(x) > 0 and since x > 0 this means log_c(log_c(x)) < 0 since we have a minus sign but this means log_c(log_c(x)) < 0 which means log_c(x) is in 0 < x < 1.

    If those conditions are satisfied we use a computer program which takes in a formula and finds what is called the root of the equation: in this particular example - we have x + log_c(log_c(x))/log_c(x) = 0 and we have to solve for x. The computer will then do its magic and get an approximate answer for x.

    Also just in case you are wondering log_c(x) = ln(x)/ln(c) where ln is the natural logarithm function which has a known formula.
     
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