3-6 Iterated functions

We can apply the iteration operator \(^\*\) used in the \(\lg^\*\) function to any monotonically increasing function \(f(n)\) over the reals. For a given constant \(c \in \mathbb R\), we define the iterated function \({f_c}^\*\) by \({f_c}^\*(n) = \min \\{i \ge 0 : f^{(i)}(n) \le c \\}\) which need not be well defined in all cases. In other words, the quantity \({f_c}^\*(n)\) is the number of iterated applications of the function \(f\) required to reduce its argument down to \(c\) or less.

For each of the following functions \(f(n)\) and constants \(c\), give as tight a bound as possible on \({f_c}^\*(n)\).

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