Is there any numerical technique for solving $a_{1}e^{x} + a_{2}e^{2x} + ... +a_{n}e^{nx}= b$, for some finite $n$?

The above expression is a polynomial of degree n and one could use a method like the Newton method. But I was wondering if there is any more efficient way of solving the above equation numerically or otherwise?

  • $\begingroup$ Your expression is not a polynomial of degree $n$, nor of any other degree. $\endgroup$ May 24, 2017 at 9:36
  • 1
    $\begingroup$ Thank you! I had posted the wrong expression! $\endgroup$
    – pkkk2012
    May 24, 2017 at 9:55
  • $\begingroup$ Newton is pretty efficient. $\endgroup$ May 24, 2017 at 9:59

1 Answer 1


Let $y=e^x$. Then $a_{1}e^{x} + a_{2}e^{2x} + \cdots +a_{n}e^{nx}= b$ becomes $a_{1}y + a_{2}y^{2} + \cdots +a_{n}y^{n}= b$, a polynomial equation in $y$. Now look for positive solutions of this equation.


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