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Math Help - How to find the solution 8*x/e^x = 2

  1. #1
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    How to find the solution 8*x/e^x = 2

    With a little bit of algebra I can also write this equation as ln 4*x = x and 4*x = e^x but that doesn't help anything.
    I'm wondering wether it is possible to find the solution of this equation without using a graphing calculator and without plugging in numbers in a methodical way.
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  2. #2
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    You won't be able to solve that in terms of "elementary functions". You could, however, rewrite it as xe^{-x}= 2 so that -xe^{-x}= -2 and then, letting y= -x, ye^y= -2.

    Now, use "Lambert's W function" which is defined as "the inverse function to f(x)= xe^{x}". Taking W of both sides, W(ye^y)= y= W(-2). Since y= -x, -x= W(-2) and x= -W(-2).
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  3. #3
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    You have discovered a great oddity. Such a simple equation that has no "algebraic" solution. You must use numerical methods. There is no "closed-form" solution for such things unless you invent one. q.v. Lambert's W function.

    There are many numerical methods that will give you as precise a result as desired - some more quickly than others.
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  4. #4
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    Suppose the problem had been e^x= 2. Would you say the same thing- "You must use numerical methods. There is no "closed-form" solution for such things unless you invent one. q.v. the natural logarithm"?

    If not, how are these two problems different?
    Last edited by HallsofIvy; October 26th 2010 at 05:41 PM.
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