How do you find an analytical solution for 3^x-x=4?
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$\begingroup$ What do you mean by "analytical"? $\endgroup$– José Figueroa-O'FarrillSep 13, 2011 at 19:43
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$\begingroup$ wolframalpha.com is your friend ;) $\endgroup$– Michael KissnerSep 13, 2011 at 19:44
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2$\begingroup$ Use the LambertW function. $\endgroup$– J.C. OttemSep 13, 2011 at 19:49
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1 Answer
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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:
$W(r)=\sum_{n=1}^\infty \frac{(-1)^{n-1}n^{n-2}}{(n-1)!}r^n$