I have been working for this problem for a while:
$$\ln(x) = 3\left(1-\frac{1}{x}\right)$$
and by graphing, plugging and chugging values and rigorously doing the math, I can clearly see that one of the values that satisfy this condition is $x = 1$.
However, when I put this into wolfram alpha and Desmos, I see two answers, one that is $x = 1$ and another one that is approximately $16.801$.
It is expressed by the Lambert W Function. I solved for $x = 1$ by using the Lambert W function. I cannot find any way to solve for the latter solution: $16.801$.
Is there anyone that could elaborate for me how to solve for it? Thank you.
My method to get $x=1$:
$$\begin{align}\ln(x) = 3\left(1-\frac{1}{x}\right)\:&\Longrightarrow\:x=e^3e^{-\frac{3}{x}}\\&\Longrightarrow\:e^{-3}=\frac{1}{x}e^{-\frac{3}{x}}\\&\Longrightarrow\:-3e^{-3}=-\frac{3}{x}e^{-\frac{3}{x}}\end{align}$$
Applying Lambert W Function, which does the following: $W(ae^a) = a $:
$$-3 = -3/x \Longleftrightarrow x=1$$
Best Answer
$-3e^{-3}=(-3/x)e^{-3/x}$ as a graph
The Lambert W function has two real valued branches when x is between -1/e and 0 exclusive, which means it is not a function in this particular case because it is not 1-1. However, if you apply the other branch of the function you will get the other value ~16.801.