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I am looking for the solution of an equation just like

$$ae^{k_{1}x} - be^{k_{2}x} + c = 0$$

where $a$, $b$, $k_{1}$, $k_{2}$ are known real constants, $x$ is unknown. I already have a numerical solution but for optimization purpose and generalization I would need to have a closed formula. It would be nice to have a solution based on well known and already well implemented functions such as Lambert function, or generalized hyper-geometric . Any opinion or clue on the way to solve such equation would be very useful.

IV_
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stykkk
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  • You could set $k_2=r\cdot k_1$, where $r$ is a rational number. You will get a radical equation in $e^x$ then. – IV_ Jun 29 '21 at 14:51

3 Answers3

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\begin{align} a\exp(k_1x) - b\exp(k_2x) + c &= 0 \tag{1}\label{1} \end{align}

The expression \eqref{1} is equivalent to

\begin{align} p\,z^u-q\,z-1&=0 \tag{2}\label{2} \\ \text{for }\quad z&=\exp(k_1\,x) ,\\ u&=\frac{k_2}{k_1} ,\quad p=\frac bc ,\quad q=\frac ac . \end{align}

Unfortunately, it is known that equation of the form \eqref{2} does not have a closed-form solution, not even in terms of the Lambert W function, unless $u$ happens to be a small integer less than $5$ hence, the numerical root finding is the only option in this case.

g.kov
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Actually, there is a new special function, the Lambert-Tsallis, proposed by R. V. Ramos at "Physica A: Statistical Mechanics and its Applications 525, 164-170" which can be used to solve. After a few manipulation you can find the folowing solution:

$$x=\frac{1}{k_1(\mu -1)} \cdot ln \left\{ \frac{-q}{p(\mu-1)}\cdot W_{\frac{\mu-2}{\mu-1}} \left[ \frac{p(\mu-1)}{(-q)^{\mu}} \right]\right\}$$

To have a solution, there are some relations to be kept among the constants respect to the functions argumentos. It would be useful if you post some groups of constants. For $a=16,b=20,c=-8, k_1=ln(3)$ and $k_2=ln(2)$ one should have $x=1$. The numerical solution for Lambert-Tsallis will be $0.307558538690483 + 0.000000000000041i$ which has generated the final $x= 0.999999999999753 - 0.000000000000329i$.

K Z Nobrega
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$$ae^{k_1x}-be^{k_2x}+c=0\tag{1}$$ $x\to\ln(t)$: $$at^{k_1}-bt^{k_2}+c=0\tag{2}$$

For $a,b,c,k_1,k_2,t\in\mathbb{N}$, equation (2) is a Diophantine equation.

For algebraic $a,b,c$ together with rational $k_1,k_2$, equation (2) is related to an algebraic equation and we can use the known solution formulas and methods for algebraic equations.

For real or complex $k_1,k_2\neq 0,1$, equation (2) is in a form similar to a trinomial equation. A closed-form solution can be obtained using confluent Fox-Wright Function $\ _1\Psi_1$ therefore.

see also: How to isolate $x$ in $a^x + b^x = c$? (For use in medical statistics)

Szabó, P. G.: On the roots of the trinomial equation. Centr. Eur. J. Operat. Res. 18 (2010) (1) 97-104

Belkić, D.: All the trinomial roots, their powers and logarithms from the Lambert series, Bell polynomials and Fox–Wright function: illustration for genome multiplicity in survival of irradiated cells. J. Math. Chem. 57 (2019) 59-106

IV_
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