Is there an algebraic solution to $e^{-x/a}+e^{-x/b}=1$ ($a\neq b$, $a,b$ constants)?

Question

Is there an algebraic solution for the to find the intersection of the following two functions for values of $x\geq 0$:
$$f_1(x)=1-2e^{-x/a}=f_2(x)=-1+2e^{-x/b}$$
$a$ and $b$ are positive constants.

The equation can be simplified to:
$$e^{-x/a}+e^{-x/b}=1$$

A Plot is here:

I am searching for the $x$-value of the intersection in the second plot (this is for an inversion recovery experiment inf magnetic resonance).

If there is no algebraic solution, can you suggest a numerical algorithm for this problem?

Thanks

Answer 1

I agree with the suggestions in the comments that for most values of $a,b$ there will be no algebraic solution. As for a numerical algorithm, are you familiar with Newton's Method (often called Newton-Raphson)?

Answer 2

Why not reduce to $y^{b/a}+y=1$, then you have only a one-parameter family to solve.

The solution for $y^r+y=1$, expanded in a Taylor series near $r=1$ is
$$ \frac{1}{2} + \frac{\operatorname{ln} (2)}{4}(r - 1) - \frac{\operatorname{ln} (2)}{8} (r - 1)^{2} - \frac{\operatorname{ln} (2) \bigl(-6 + 2 \operatorname{ln} (2)^{2} - 3 \operatorname{ln} (2)\bigr)}{96} (r - 1)^{3} + \frac{\operatorname{ln} (2) \bigl(2 \operatorname{ln} (2) + 1\bigr) \bigl(\operatorname{ln} (2) - 2\bigr)}{64} (r - 1)^{4} + \operatorname{O} \bigl((r - 1)^{5}\bigr) $$