Is there any numerical technique for solving $a_{1}\exp(x) + a_{2}\exp(2x) + ... +a_{n}\exp(nx)$, for some finite n?

Question

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?

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.