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

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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:

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

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