Answer: deal with iterations of the logarithm function.

Chapter 5, Problem 62E

(choose chapter or problem)

Find the largest integer \(n\) such that \(\log ^{*} n=5\). Determine the number of decimal digits in this number.

Exercises 63-65 deal with values of iterated functions. Suppose that \(f(n)\) is a function from the set of real numbers, or positive real numbers, or some other set of real numbers, to the set of real numbers such that \(f(n)\) is monotonically increasing [that is, \(f(n)<f(m)\) when \(n<m\) ) and \(f(n)<n\) for all \(n\) in the domain of \(f .]\) The function \(f^{(k)}(n)\) is defined recursively by

                         \(f^{(k)}(n)= \begin{cases}n & \text { if } k=0 \\ f\left(f^{(k-1)}(n)\right) & \text { if } k>0\end{cases}\)

Furthermore, let \(c\) be a positive real number. The iterated function \(f_{c}^{*}\) is the number of iterations of \(f\) required to reduce its argument to \(c\) or less, so \(f_{c}^{*}(n)\) is the smallest nonnegative integer \(k\) such that \(f^{k}(n) \leq c\).

Equation Transcription:

Text Transcription:

n

lof(n)g*⁡n=5

f(n)<n

f(n)<f(m)

n<m

f^(k)(n)

f^(k)(n)={n  if k=0 f(f^(k-1)(n))  if k>0

f_c^*

f^k(n) < or = c