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How do you find an analytical solution for 3^x-x=4?

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closed as off topic by J.C. Ottem, Denis Serre, quid, Daniel Litt, Will Jagy Sep 13 '11 at 21:21

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What do you mean by "analytical"? –  José Figueroa-O'Farrill Sep 13 '11 at 19:43
    
wolframalpha.com is your friend ;) –  Michael Kissner Sep 13 '11 at 19:44
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Use the LambertW function. –  J.C. Ottem Sep 13 '11 at 19:49
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1 Answer 1

If you want, you can use Lambert's W function. Let $z:=x+4$, so that the equation becomes $3^{z-4}=z$, so that $-3^{-4}=-z3^{-z}$, which upon inverting with Lambert's $W$ gives:

$x=-\frac{W(-\ln(3)3^{-4})}{\ln(3)}-4$

In particular, $W(r)$ has a nice taylor expansion:

$W(r)=\sum_{n=1}^\infty \frac{(-1)^{n-1}n^{n-2}}{(n-1)!}r^n$

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