Evaluating magnitudes of non-algebraic numbers

[espen180]

Just like it is possible to show that $$e^\pi > \pi^e$$, is it possible to show that

$$\sqrt{2}^{\sqrt{2}} > \frac{1+\sqrt{5}}{2}$$

or

$$\sqrt{2}^{\sqrt{2}} < \sqrt{3}$$

 

[mathman]

Just like it is possible to show that $$e^\pi > \pi^e$$, is it possible to show that

$$\sqrt{2}^{\sqrt{2}} > \frac{1+\sqrt{5}}{2}$$

or

$$\sqrt{2}^{\sqrt{2}} < \sqrt{3}$$

Since all these numbers are well defined, it is possible, although at times it might require a little work.