Question

Find the value of \({\log ^*}n\) for these values of n.

1. \(2\)
2. \(4\)
3. \(8\)
4. \(16\)
5. \(256\)
6. \(65536\)
7. \({2^{2048}}\)

Short answer

1. \({\log ^ * }2 = 1\)
2. \({\log ^ * }4 = 2\)
3. \({\log ^ * }8 = 3\)
4. \({\log ^ * }16 = 3\)
5. \({\log ^ * }256 = 4\)
6. \({\log ^ * }65536 = 4\)
7. \({\log ^ * }{2^{2048}} = 5\)

Step-by-step solution

 Step 1: Describe the given information and formulas for logarithm

It is given that,

\({\log ^{\left( k \right)}}n = \;\left\{ \begin{aligned}{l}n\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{if}}\;k = 0\\\log \left( {{{\log }^{\left( {k - 1} \right)}}n} \right)\;\;\;\;\;\;\;\;{\rm{if lo}}{{\rm{g}}^{\left( {k - 1} \right)}}n{\rm{ is defined and positive}}\\{\rm{undefined}}\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{otherwise}}{\rm{.}}\end{aligned} \right.\)

Where,\({\log ^*}n\)is the smallest nonnegative integer ksuch that\({\log ^{\left( k \right)}}n \le 1\).

Determine the value of \({\log ^*}n\) for \(n = 2\)(a)

Determine \({\log ^{\left( k \right)}}n\) for increasing values of \(k\) until a value of at most \(1\) is obtained. Use base \(2\) for the logarithm.

\(\begin{aligned}{c}{\log ^{\left( 0 \right)}}2 = 2\\{\log ^{\left( 1 \right)}}2 = \log \left( {{{\log }^{\left( 0 \right)}}2} \right)\\ = \log \left( 2 \right)\\ = 1\end{aligned}\)

\({\log ^*}n\)is the smallest nonnegative integer ksuch that \({\log ^{\left( k \right)}}n \le 1\).

\({\log ^ * }2 = 1\)

Therefore, the value of \({\log ^ * }2\) is \(1\).

Determine the value of \({\log ^*}n\) for \(n = 4\)(b)

Determine \({\log ^{\left( k \right)}}n\) for increasing values of \(k\) until a value of at most \(1\) is obtained. Use base \(2\) for the logarithm.

\(\begin{aligned}{c}{\log ^{\left( 0 \right)}}4 = 4\\{\log ^{\left( 1 \right)}}4 = \log \left( {{{\log }^{\left( 0 \right)}}4} \right)\\ = \log \left( 4 \right)\\ = 2\end{aligned}\)

And,

\(\begin{aligned}{c}{\log ^{\left( 2 \right)}}4 = \log \left( {{{\log }^{\left( 1 \right)}}4} \right)\\ = \log \left( 2 \right)\\ = 1\end{aligned}\)

\({\log ^*}n\)is the smallest nonnegative integer ksuch that \({\log ^{\left( k \right)}}n \le 1\).

\({\log ^ * }4 = 2\)

Therefore, the value of \({\log ^ * }4\) is \(2\).

Determine the value of \({\log ^*}n\) for \(n = 8\)(c)

Determine \({\log ^{\left( k \right)}}n\) for increasing values of \(k\) until a value of at most \(1\) is obtained. Use base \(2\) for the logarithm.

\(\begin{aligned}{c}{\log ^{\left( 0 \right)}}8 = 8\\{\log ^{\left( 1 \right)}}8 = \log \left( {{{\log }^{\left( 0 \right)}}8} \right)\\ = \log \left( 8 \right)\\ = 3\end{aligned}\)

And,

\(\begin{aligned}{c}{\log ^{\left( 2 \right)}}8 = \log \left( {{{\log }^{\left( 1 \right)}}8} \right)\\ = \log \left( 3 \right)\\ \approx 1.5850\end{aligned}\)

And,

\(\begin{aligned}{c}{\log ^{\left( 3 \right)}}8 = \log \left( {{{\log }^{\left( 2 \right)}}8} \right)\\ = \log \left( {\log \left( 3 \right)} \right)\\ \approx 0.6644\end{aligned}\)

\({\log ^*}n\)is the smallest nonnegative integer ksuch that \({\log ^{\left( k \right)}}n \le 1\).

\({\log ^ * }8 = 3\)

Therefore, the value of \({\log ^ * }8\) is \(3\).

Determine the value of \({\log ^*}n\) for \(n = 16\)(d)

Determine \({\log ^{\left( k \right)}}n\) for increasing values of \(k\) until a value of at most \(1\) is obtained. Use base \(2\) for the logarithm.

\(\begin{aligned}{c}{\log ^{\left( 0 \right)}}16 = 16\\{\log ^{\left( 1 \right)}}16 = \log \left( {{{\log }^{\left( 0 \right)}}16} \right)\\ = \log \left( {16} \right)\\ = 4\end{aligned}\)

And,

\(\begin{aligned}{c}{\log ^{\left( 2 \right)}}16 = \log \left( {{{\log }^{\left( 1 \right)}}16} \right)\\ = \log \left( 4 \right)\\ = 2\end{aligned}\)

And,

\(\begin{aligned}{c}{\log ^{\left( 3 \right)}}16 = \log \left( {{{\log }^{\left( 2 \right)}}16} \right)\\ = \log \left( 2 \right)\\ = 1\end{aligned}\)

\({\log ^*}n\)is the smallest nonnegative integer ksuch that \({\log ^{\left( k \right)}}n \le 1\).

\({\log ^ * }16 = 3\)

Therefore, the value of \({\log ^ * }16\) is \(3\).

Determine the value of \({\log ^*}n\) for \(n = 256\)(e)

Determine \({\log ^{\left( k \right)}}n\) for increasing values of \(k\) until a value of at most \(1\) is obtained. Use base \(2\) for the logarithm.

\(\begin{aligned}{c}{\log ^{\left( 0 \right)}}256 = 256\\{\log ^{\left( 1 \right)}}256 = \log \left( {{{\log }^{\left( 0 \right)}}256} \right)\\ = \log \left( {256} \right)\\ = 8\end{aligned}\)

And,

\(\begin{aligned}{c}{\log ^{\left( 2 \right)}}256 = \log \left( {{{\log }^{\left( 1 \right)}}256} \right)\\ = \log \left( 8 \right)\\ = 3\end{aligned}\)

And,

\(\begin{aligned}{c}{\log ^{\left( 3 \right)}}256 = \log \left( {{{\log }^{\left( 2 \right)}}256} \right)\\ = \log \left( 3 \right)\\ \approx 1.5850\end{aligned}\)

And,

\(\begin{aligned}{c}{\log ^{\left( 4 \right)}}256 = \log \left( {{{\log }^{\left( 3 \right)}}256} \right)\\ = \log \left( {\log \left( 3 \right)} \right)\\ \approx 0.6644\end{aligned}\)

\({\log ^*}n\)is the smallest nonnegative integer ksuch that \({\log ^{\left( k \right)}}n \le 1\).

\({\log ^ * }256 = 4\)

Therefore, the value of \({\log ^ * }256\) is \(4\).

Determine the value of \({\log ^*}n\) for \(n = 65536\)(f)

Determine \({\log ^{\left( k \right)}}n\) for increasing values of \(k\) until a value of at most \(1\) is obtained. Use base \(2\) for the logarithm.

\(\begin{aligned}{c}{\log ^{\left( 0 \right)}}65536 = 65536\\{\log ^{\left( 1 \right)}}65536 = \log \left( {{{\log }^{\left( 0 \right)}}65536} \right)\\ = \log \left( {65536} \right)\\ = 16\end{aligned}\)

And,

\(\begin{aligned}{c}{\log ^{\left( 2 \right)}}65536 = \log \left( {{{\log }^{\left( 1 \right)}}65536} \right)\\ = \log \left( {16} \right)\\ = 4\end{aligned}\)

And,

\(\begin{aligned}{c}{\log ^{\left( 3 \right)}}65536 = \log \left( {{{\log }^{\left( 2 \right)}}65536} \right)\\ = \log \left( 4 \right)\\ = 2\end{aligned}\)

And,

\(\begin{aligned}{c}{\log ^{\left( 4 \right)}}65536 = \log \left( {{{\log }^{\left( 3 \right)}}65536} \right)\\ = \log \left( 2 \right)\\ = 1\end{aligned}\)

\({\log ^*}n\)is the smallest nonnegative integer ksuch that \({\log ^{\left( k \right)}}n \le 1\).

\({\log ^ * }65536 = 4\)

Therefore, the value of \({\log ^ * }65536\) is \(4\).

Determine the value of \({\log ^*}n\) for \(n = {2^{2048}}\)(g)

Determine \({\log ^{\left( k \right)}}n\) for increasing values of \(k\) until a value of at most \(1\) is obtained. Use base \(2\) for the logarithm.

\(\begin{aligned}{c}{\log ^{\left( 0 \right)}}{2^{2048}} = {2^{2048}}\\{\log ^{\left( 1 \right)}}{2^{2048}} = \log \left( {{{\log }^{\left( 0 \right)}}{2^{2048}}} \right)\\ = \log \left( {{2^{2048}}} \right)\\ = 2048\end{aligned}\)

And,

\(\begin{aligned}{c}{\log ^{\left( 2 \right)}}{2^{2048}} = \log \left( {{{\log }^{\left( 1 \right)}}{2^{2048}}} \right)\\ = \log \left( {2048} \right)\\ = 11\end{aligned}\)

And,

\(\begin{aligned}{c}{\log ^{\left( 3 \right)}}{2^{2048}} = \log \left( {{{\log }^{\left( 2 \right)}}{2^{2048}}} \right)\\ = \log \left( {11} \right)\\ \approx 3.4594\end{aligned}\)

And,

\(\begin{aligned}{c}{\log ^{\left( 4 \right)}}{2^{2048}} = \log \left( {{{\log }^{\left( 3 \right)}}{2^{2048}}} \right)\\ = \log \left( {\log \left( {11} \right)} \right)\\ \approx 1.7905\end{aligned}\)

And,

\(\begin{aligned}{c}{\log ^{\left( 5 \right)}}{2^{2048}} = \log \left( {{{\log }^{\left( 4 \right)}}{2^{2048}}} \right)\\ = \log \left( {\log \left( {\log \left( {11} \right)} \right)} \right)\\ \approx 0.8404\end{aligned}\)

\({\log ^*}n\)is the smallest nonnegative integer ksuch that \({\log ^{\left( k \right)}}n \le 1\).

\({\log ^ * }{2^{2048}} = 5\)

Therefore, the value of \({\log ^ * }{2^{2048}}\) is \(5\).