Exponential Equations - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-06-19T15:40:15Z http://mathoverflow.net/feeds/question/75345 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/75345/exponential-equations Exponential Equations Martin Arnolf 2011-09-13T19:36:44Z 2011-09-13T19:59:17Z <p>How do you find an analytical solution for 3^x-x=4?</p> http://mathoverflow.net/questions/75345/exponential-equations/75348#75348 Answer by Alex R. for Exponential Equations Alex R. 2011-09-13T19:59:17Z 2011-09-13T19:59:17Z <p>If you want, you can use <a href="http://en.wikipedia.org/wiki/Lambert_W_function" rel="nofollow">Lambert's W function</a>. 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:</p> <p>$x=-\frac{W(-\ln(3)3^{-4})}{\ln(3)}-4$</p> <p>In particular, $W(r)$ has a nice taylor expansion:</p> <p>$W(r)=\sum_{n=1}^\infty \frac{(-1)^{n-1}n^{n-2}}{(n-1)!}r^n$</p>