Solving (or approximating a solution of) $c = (1+a)^x - (1-a)^x$

Question

I am looking to solve the following equation:

$$(1+a)^x - (1-a)^x = c$$

I know that $c ≥ 1$, $a \in [0; 1]$, and I am looking for the only solution where $x ≥ 0$.

I am aware that this is a complex equation and there might not exist a closed form for the solution. If there is one, even using "funkier" functions, I'd be interested in knowing it. If not, which algorithm would be efficient here to quickly approximate the solution?

Answer 1

Since $c\ge 1$, I think it's profitable to think of the $-(1-a)^x$ term as "small", so that the $(1+a)^x$ term controls the behavior of the solution. For example:

1. We can get an easy lower bound using only first term: $x\ge \log c/\log(1+a)$, since $c\le (1+a)^x$.

2. Using (1), $(1-a)^x\le (1-a)^{\log c/\log(1+a)}$, which means $(1+a)^x\le c+(1-a)^{\log c/\log(1+a)}$. This gives an upper bound for $x$: $$ x\le \frac{\log\left[c+(1-a)^{\log c/\log(1+a)}\right]}{\log(1+a)}. $$ Obviously, these bounds are sharp in the cases $a=1$ and $\lim a=0$.

Answer 2

$$(1+a)^x-(1-a)^x=c$$

We see, your equation is a polynomial equation of more than one algebraically independent monomials ($(1+a)^x,(1-a)^x$), without a univariate factor. We therefore don't know how to rearrange the equation for $x$ by applying only finite numbers of elementary functions (elementary operations) we can read from the equation.

$$e^{\ln(1+a)x}-e^{\ln(1-a)x}=c$$ $x\to\ln(t),a_1=\ln(1+a),a_2=1-a$: $$t^{a1}-t^{a2}=c$$ $$t^{a1}-t^{a2}-c=0$$

We see, this equation is a trinomial equation with real exponents. We can transform it to a form like in equation 8.1 of [Belkic 2019]. Solutions in terms of Bell polynomials, Pochhammer symbols or confluent Fox-Wright Function $\ _1\Psi_1$ can be obtained therefore.

[Belkić 2019] 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

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