Show there is exactly one such that
for
and
an odd, positive integer.
Proof. Let , and let
with
. Then,
(for odd
) so
. Since
and
, by the Intermediate Value Theorem, we know
takes every value between
and 0 for some
. Thus, we know there exists
such that
(since
). This implies
for some
.
We know this solution is unique since is strictly increasing on the whole real line for odd
Just a slight correction. For odd n (including 1) c^n <= c.