2

What is the solution for $a^x + bx = c$? Also, can anyone refer me to a good article/book/etc that covers general solution methods for exponential functions?

Thanks,

Zev Chonoles
  • 129,973
Larry Lee
  • 23
  • There is no solution for such equations in general (with elementary operations/functions). Equations of this type are transcendental and must be solved numerically. Even an equation as simple as $e^x = x$ cannot be solved with elementary operations/functions. – Cameron Williams Jun 02 '13 at 02:23
  • 1
    Well, there's a name for the function needed to solve this equation -- the Lambert W function -- and there are books and so forth written about it, so it's a bit forward to say that all you can do is solve it numerically. For sure it's not an elementary function, but its properties are well understood, and there are nice diffeqs, power series, etc. for it. – Daniel McLaury Jun 02 '13 at 02:26
  • Thanks for the prompt response. I'll take some time to look up the Lambert W function. Do either of you have a good reference for transcendental functions that you could point me too? A reference that provides conditions for recognizing transcendental functions would be ideal. Thanks again. – Larry Lee Jun 02 '13 at 02:36

1 Answers1

1

This can be solved in terms of the Lambert W function, which is defined as the solution for $w$ to $z = w e^w$. I think you'd read about the Lambert W function in a book or chapter with a title like "special functions" that just listed a lot of these various functions that can be used in weird circumstances like this, but I've never actually read one myself because I've never had any need to.

http://en.wikipedia.org/wiki/Lambert_W_function

Daniel McLaury
  • 24,656