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Math Help - Inverse Function of Natural Logs

  1. #1
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    Inverse Function of Natural Logs

    Hi everyone... I've been working on this problem for about a week and I just can't seem to get it. I got the rest of the problems in my homework done but I cant get this one.

    A function f (x) is given.
    f
    (x) = ln(ln(ln 8x))

    Find the inverse function of f.
    f -1 (x) =

    I don't know where to go from here? Any help would be great. Thank you so much!

    (PS, I know x = e^y if that helps at all! I dont know where to go now though)
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  2. #2
    Senior Member
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    Hi ZBomber

    Of course it helps a lot to know x = e^y.

    y = ln(ln(ln 8x))

    The inverse is :
    x= ln(ln(ln 8y)) , now isolate y

    e^x=ln(ln 8y)

    ln(8y) = .... , if you confuse, just let e^x = a
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  3. #3
    Junior Member woof's Avatar
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    It's all based on the fact that e^{ln(whatever)}=whatever

    y=ln(ln(ln(8x))), the inverse is the reflection: x=ln(ln(ln(8y))), and solve for y...by applying e^{?} to both sides

    e^x=e^{ln(ln(ln(8y)))}

    e^x={ln(ln(8y))}

    e^{e^x}=e^{ln(ln(8y))}

    e^{e^x}={ln(8y)}

    e^{e^{e^x}}=e^{ln(8y)}

    e^{e^{e^x}}=8y, then.....


    y={1\over8}e^{e^{e^x}} and therefore f^{-1}(x)={1\over8}e^{e^{e^x}}
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  4. #4
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    Mar 2008
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    Thank you both so much!

    Strangely, I was headed towards the right answer, but I thought it looked way too complicated to be correct so I stopped before I got to the end. I guess I was wrong in that regard.

    Again, thank you both!
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