Deal with iterations of the logarithm function. Let log n...

Deal with iterations of the logarithm function. Let log $n$ denote the logarithm of $n$ to the base $2,$ as usual. The function $\log ^{(k)} n$ is defined recursively by $$\log ^{(k)} n=\left\{\begin{array}{cl}{n} & {\text { if } k=0} \\ {\log \left(\log ^{(k-1)} n\right)} & {\text { if } \log ^{(k-1)} n \text { is defined }} \\ {} & {\text { and positive }} \\ {\text { undefined }} & {\text { otherwise. }}\end{array}\right.$$ The iterated logarithm is the function log" $n$ whose value at $n$ is the smallest nonnegative integer $k$ such that $\log ^{(k)} n \leq 1$ Find the value of $\log ^{*} n$ for these values of $n$ a) 2 b) 4 c) 8 d) 16 e) 256 f) 65536 g) $2^{2048}$

Deal with iterations of the logarithm function. Let log $n$ denote the logarithm of $n$ to the base $2,$ as usual. The function $\log ^{(k)} n$ is defined recursively by
$$\log ^{(k)} n=\left\{\begin{array}{cl}{n} & {\text { if } k=0} \\ {\log \left(\log ^{(k-1)} n\right)} & {\text { if } \log ^{(k-1)} n \text { is defined }} \\ {} & {\text { and positive }} \\ {\text { undefined }} & {\text { otherwise. }}\end{array}\right.$$
The iterated logarithm is the function log" $n$ whose value at $n$ is the smallest nonnegative integer $k$ such that $\log ^{(k)} n \leq 1$
Find the value of $\log ^{*} n$ for these values of $n$
a) 2
b) 4
c) 8
d) 16
e) 256
f) 65536
g) $2^{2048}$

Key Concepts

Logarithm Function

The logarithm function, particularly to the base 2 in this context, is the inverse of the exponential function with the same base. It measures how many times one must multiply the base to obtain a given number, serving as a fundamental tool in various mathematical analyses including complexity and growth rates.

Recursion and Recursive Definitions

A recursive definition specifies an object in terms of simpler instances of itself, using base cases and recursive steps. In the context of the iterated logarithm, the function is defined recursively, starting from an initial value and repeatedly applying the logarithm operation, which is crucial for understanding how repeated operations affect values.

Iterated Logarithm

The iterated logarithm involves the repeated application of the logarithm function. It provides insight into the number of times the logarithm must be applied before the result is reduced to a certain bound, making it useful in theoretical computer science and complexity analysis.

Log-Star Function

The log-star function (denoted as log* n) represents the smallest number of times the logarithm function must be applied to a number before the result is less than or equal to a given threshold (commonly 1). This function grows extremely slowly, and is important in analyzing the complexity of algorithms and data structures where iterative logarithmic reductions are involved.

Transcript

00:08 We are asked to find the values, iterations.

00:21 So we have, first of all, that log n is denotes log base 2, and we have that log k of n is defined, n, if k is equal to 0, and log of log of k minus 1, if k, or if log of k minus 1 is defined, and of course it's going to be undefined, otherwise.

01:53 So we have to find the values of this function.

01:59 So in part a you want to find log 2 of 16.

02:15 We have that in this case k is equal to 2.

02:31 So this is going to be equal to the log 1 of 16 as long as defined greater than 0 and this is equal to the log 0 of 16.

03:06 This is defined.

03:06 And the log 0 of 16.

03:14 This is defined.

03:21 And the log 0.

03:23 0 of 16 is simply this is going to be the base 2 logarithm of 16.

03:37 So the log of in this case is 4, it's 2 to 4 equals 16.

03:45 The log of 4 is 2.

03:50 Part a we get 2.

03:53 Part b we're asked to find log 3 of 256.

04:11 So this is equal 2 by definition.

04:14 The log of log 2 56, as long as 1.

04:21 This second part is defined, which is then equal to the log of the log, log 1 of 256, as long as log 1 of 2 .56 is well defined, which is equal to the log, the log of the log of the log of the log 0, 256, as long as log 0 of 256 is well defined.

04:59 And in fact, it is well defined, and this is equal to the log, and then log 0.

05:09 0 to 256 is by definition 256.

05:25 So call that 2 the 8th is 256.

05:31 So this is actually equal to the log of the log of 8...

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