Daniel Wrote:bo198214 Wrote:Is Andrew's solution for bases\( b\le e^{1/e} \) identical with the solution obtained by regular development at the lower fixed point?Can you clarify this comment please?
If I correctly understood you then your approach was to develop the function \( b^x=\exp_b(x) \) at a fixed point and then regularly iterate the function there \( \exp_b^{\circ t}(x) \) and then define tetration as \( {}^tb=\exp_b^{\circ t}(1) \). It quite looks (as I numerically verified in this post) as if this yields the same result as Andrew's approach (of course only for bases \( b\le\eta=e^{1/e} \) because only there exists a real fixed point at all. The solution can also only applied to the lower fixed point (of the two fixed points of \( b^x \) for \( b<\eta \)), because the other fixed point is not reachable from 1.)