Exercise
6
[20W] Prerequisites:[20V].Difficulty:*.Having fixed \(๐ผ {\gt}1\), we define, for \(xโโ\),
\[ ๐ผ ^ x=\sup \{ ๐ผ^ p : pโโ, pโค x\} \quad ; \]
show that:
this is a good definition (i.e. that the set on the right is bounded above and not empty).
Iff \(x\) is rational then \(๐ผ ^ x\) (as above defined) coincides with the definition in the previous exercise [20V].
show that \(xโฆ ๐ผ^ x\) is strictly increasing.
Show that
\[ ๐ผ^{x}๐ผ^ y=๐ผ^{x+y}\quad ,\quad (๐ผ^ x)^ y=๐ผ^{(xy)}\quad . \]
See also the exercise [21N].