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.