Is it possible to inverse a sum of exponents

Question

I have a problem, I need to inverse a sum of exponents. Is it possible?

I have this function

$y = 0.84826731\times e^{-1.10973369x} + 0.17939312\times e^{-0.1902204x} + 0.02965983\times e^{-0.02028059x}$

and I would like to get to

$$x = f(y)$$

The ranges in which I need x = f(y), is y between (0,1), which results in x in the range of ~(0,100)

It does not have to be 100% correct. 99.9% will also be ok.

Thanks.

Answer 1

Hint: use taylor series to approximate $y$ in form of one exponential and then use logarithm both side to get inverse.

Answer 2

If $y$ is very large, then $x$ must be negative, and the first term is dominant. So $x\approx\frac{-1}{1.10973369}\ln\mathopen{}\left(\frac{y}{0.84826731}\right)\mathclose{}$.

If $y$ is very small, then $x$ must be positive, and the last term is dominant. So $x\approx\frac{-1}{0.02028059}\ln\mathopen{}\left(\frac{y}{0.02965983}\right)\mathclose{}$.

I guess you can do some numerical experiments to see how large is large enough and how small is small enough for these approximations. A purely visual inspection of some graphs suggests to me that these are decent for $y>10$ and $y<0.02$. For $y$ in the middle region, you have other approximation options like numerical solution from Newton's method.