5
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Canadian economist Mike Moffat asks on Twitter:

Math nerd Q: Is there a way to solve $e^x + x = 5$ for $x$, without using a numerical method?

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    So, based on the answers given, "NO".2011-08-03
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    I guess it depends on what you mean by 'numerical method'. Sure, you need to use a numerical method to compute the W function (Newton's method works well). But you need to use a numerical method to compute values of the exponential function, sine function or even the square root function, and this isn't really so different.2011-08-03
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    If you want a closed form expression for the solution in terms of elementary functions, then the answer is *no*.2011-08-03
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    @Chris Taylor: Hurrah for the comment about needing a numerical method for exponential, sine, square root. There are a number of questions on this site for which this should be part of the answer.2011-08-03
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    @lhf: technically, you are incorrect. Every constant function is elementary, and some constant function has this value... However, the function $x = g(y)$ solving $e^x+x=y$ is, indeed, non-elementary.2011-08-03
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    @GEdgar, you're right, that's what I meant. It *may* happen that the solution for $y=5$ has an elementary form, though I doubt it, 5 not looking special in any way.2011-08-03
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    @Chris: Yes! YES! That's *precisely* what annoys me about people who say "I don't want to use a numerical method"; even using the humble square root function of your computing environment is a bleeding numerical method! You said it, dude!2011-08-04

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