MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question

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

Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here.If this question can be reworded to fit the rules in the help center, please edit the question.

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

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:


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

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

share|cite|improve this answer

Not the answer you're looking for? Browse other questions tagged or ask your own question.