How do you find an analytical solution for 3^xx=4?
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If you want, you can use Lambert's W function. Let $z:=x+4$, so that the equation becomes $3^{z4}=z$, 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: $W(r)=\sum_{n=1}^\infty \frac{(1)^{n1}n^{n2}}{(n1)!}r^n$ 

