Unique Exponential Equation

Question

I solved the following equation by inspection and also by graphing (Desmos). x = 1. Is it possible to put this equation into Lambert form? I tried but could not do it. Thanks. $$ 2^x = x + 1/x $$

Answer 1

$$2^x=x+\frac{1}{x}$$

$$e^{\ln(2)x}=x+\frac{1}{x}$$

$$\frac{x}{x^2+1}e^{\ln(2)x}=1$$

One real solution is $x=1$.

We see, your equation cannot be solved in terms of Lambert W. But it is solvable in terms of Generalized Lambert W:

$$x=\frac{1}{\ln(2)}W\left(^{\ \ \ \ \ \ \ \ \ \ \ 0}_{-\ln(2)i,\ln(2)i};\frac{1}{\ln(2)}\right)=-\frac{1}{\ln(2)}W\left(^{-\ln(2)i,\ln(2)i}_{\ \ \ \ \ \ \ \ \ \ \ 0} ;\ln(2)\right)$$

$-$ see the references below.
$\ $

[Mezö 2017] Mezö, I.: On the structure of the solution set of a generalized Euler-Lambert equation. J. Math. Anal. Appl. 455 (2017) (1) 538-553

[Mezö/Baricz 2017] Mezö, I.; Baricz, A.: On the generalization of the Lambert W function. Transact. Amer. Math. Soc. 369 (2017) (11) 7917–7934 (On the generalization of the Lambert W function with applications in theoretical physics. 2015)

[Castle 2018] Castle, P.: Taylor series for generalized Lambert W functions. 2018