TITLE: Why does iterating in different ways produce different solutions?
QUESTION [6 upvotes]: Recently, I came across the following equation:
$$2^x=4x$$
To solve it, I decided to iterate. Firstly I stated:
$$x_{n+1}=\frac{2^{x_n}}{4},x_0=1$$ and found a solution of $x\approx 0.3099069324$. 
Then I again rearranged it to:
$$x_{n+1}=\log_{2}({4x_n}), x_0=1,$$
and achieved the solution $x=4$.
Despite trying many values, I was unable to get these iterations to find the other solution. What I would like to know is: what is the reasoning behind this?
I have seen a similar thing happen in the case of other iterations, where I iterate for a solution in different ways and get two different (correct) solutions - and it has perplexed me.

REPLY [3 votes]: An "exact" solution may be obtained with use of the Lambert W-function which is defined by $W_{0}(x) e^{W_{0}(x)} = x$ for $x \geq -1$. This is seen by the following:
\begin{align}
2^{x} &= 4x \\
4 x &= e^{x \, \ln2} \\
(- x \, \ln2) \, e^{- x \, \ln2} &= - \frac{\ln2}{4} \\
- x \, \ln2 &= W_{0}\left( - \frac{\ln2}{4} \right) \\
x &= - \frac{1}{\ln2} \, W_{0}\left( - \frac{\ln2}{4} \right).
\end{align}
By calculation this gives a value of $x \approx .30990693238\cdots$.