Golden ratio

Solve an equation involving the Golden Ratio phi where the unknown occurs as a power of phi.

Problem



The 'divina proporzione' or golden ratio, represented by the Greek letter phi, is derived from the equation below where $a$ and $b$ are parts of a line.

Image
Golden Ratio
$a+b:a=a:b$


i.e. $ \frac{a+b}{a}=\frac{a}{b}=\Phi\ \quad $(phi)


If $b = 1$ show that $\Phi = a = (\sqrt 5 + 1 )/2 = 1.618034...$.


In the following equation what does $x$ equal?


$$\Phi^{\left(\Phi^x-\frac{x-1}{\Phi}\right)}-\frac{1}{\Phi}=x$$