Question

Find these values.

$\begin{array}{r}{\mathbf{log}}^{\mathbf{\left(}\mathbf{2}\mathbf{\right)}}\mathbf{}\mathbf{16}\\ {\mathbf{log}}^{\mathbf{\left(}\mathbf{3}\mathbf{\right)}}\mathbf{}\mathbf{256}\\ {\mathbf{log}}^{\mathbf{\left(}\mathbf{3}\mathbf{\right)}}\mathbf{}{\mathbf{2}}^{\mathbf{65536}}\\ {\mathbf{log}}^{\mathbf{\left(}\mathbf{4}\mathbf{\right)}}\mathbf{}{\mathbf{2}}^{{\mathbf{2}}^{\mathbf{65536}}}\end{array}$

Short answer

(a) The value of ${\log }^{\left(2\right)}16$is 2.

(b) The value of${\log }^{\left(3\right)}256$ is.

(c) The value of ${\log }^{\left(3\right)}{2}^{65536}$is 4.

(d) The value of ${\log }^{\left(4\right)}{2}^{65536}$is 4.

Step-by-step solution

Define multiplicative cyclic group

The logarithm that refer to the multiplicative cyclic group are the discrete logarithm. Consider G is the group with the generator G, then by the definition for each element of G is ${\mathbf{g}}^{\mathbf{x}}$, here is the specific value.

The definition of iterated logarithm is as follows

${\log }^{\left(k\right)}n=\left\{\begin{array}{c}n\text{if}k=0\\ \log \left({\log }^{(k-1)}n\right)\text{if}{\log }^{(k-1)}n\text{is defined and positive}\\ \text{undefined otherwise}\end{array}\right\}$

Consider the log to the base 2

Consider log2 = 1

(a) Find  log(2)16

Determine the value of ${\log }^{\left(2\right)}16$as follows:

$\begin{array}{r}{\log }^{\left(2\right)}16=\log \left({\log }^{\left(1\right)}16\right)\\ =\log (\log (\log \left(0\right)16\left)\right)\\ =\log (\log 16)\\ =\log \left(\log 2^4\right)\\ \mathrm{Substitute}\mathrm{the}\mathrm{value}\mathrm{and}\mathrm{solve}\mathrm{as}:\\ \log \left(2\right)16=\log (4\log 2)\\ =\log 2^2\\ =2\end{array}$

Hence, the value is 2.

(b) Find  log(3)256

Determine the value of ${\log }^{\left(3\right)}256$as follows:

$\begin{array}{r}{\log }^{\left(3\right)}256=\log (\log (3\left)256\right)\\ =(\log (\log \left(3\right)256\left)\right)\\ =\log (\log (\log (\log (0\left)\right)\left)\right)\\ =\log (\log (\log \left(256\right)\left)\right)\\ \mathrm{Substitute}\mathrm{the}\mathrm{value}\mathrm{and}\mathrm{solve}\mathrm{as}:\\ {\log }^{\left(3\right)}256=\log \left(\log \left(\log 2^8\right)\right)\\ =\log (\log (8\log 2\left)\right)\\ =\log \left(\log 2^3\right)\\ =\log 3\end{array}$

Hence, the value is log 3.

(c) Find  log(3)265536

Determine the value of ${\log }^{\left(3\right)}{2}^{65536}$ as follows:

$\begin{array}{r}{\log }^{\left(3\right)}{2}^{65536}=\log \left(\log \left(2\right)2^{65536}\right)\\ =\log \left(\log \left({\log }^{\left(\mathrm{f}\right)}{2}^{65536}\right)\right)\\ =\log \left(\log \left(\log \left({\log }^{\left(0\right)}{2}^{65536}\right)\right)\right)\\ =\log \left(\log \left(\log \left(2^{65536}\right)\right)\right)\end{array}$

Substitute the value and solve as:

$\begin{array}{r}{\log }^{\left(3\right)}{2}^{65536}=\log (\log (65536\log 2\left)\right)\\ =\log \left(\log \left(2^{16}\right)\right)\\ =\log (16\log (2\left)\right)\\ =\log \left(2^4\right)\\ =4\\ \mathrm{Solve}\mathrm{further}\mathrm{as}:\\ \mid {\log }^{\left(3\right)}{2}^{65536}=4\end{array}$

Hence, the value is 4.

(d) Find  log(4)2265536

$\mathrm{Determine}\mathrm{the}\mathrm{value}\mathrm{of}{\log }^{\left(4\right)}{2}^{2^{65536}}\mathrm{as}\mathrm{follows}:$

$\begin{array}{r}{\log }^{\left(4\right)}{2}^{2^{65536}}=\log \left(\log \left(3\right)2^{2^{65536}}\right)\\ =\log \left(\log \left(\log \left(2\right)2^{2^{65536}}\right)\right)\\ =\log \left(\log \left(\log \left(\log {}^{\left(1\right)}{2}^{2^{65536}}\right)\right)\right)\\ =\log \left(\log \left(\log \left(\log \left(\log {}^{\left(0\right)}{2}^{2^{65556}}\right)\right)\right)\right)\\ \mathrm{Solve}\mathrm{further}\mathrm{as}:\\ {\log }^{\left(4\right)}{2}^{2^{65536}}=\log \left(\log \left(\log \left(\log \left(2^{2^{65556}}\right)\right)\right)\right)\\ =\log \left(\log \left(\log \left(2^{65536}\right)\right)\right)\\ =\log (\log (65536\left)\right)\\ =\log \left(\log \left(2^{16}\right)\right)\end{array}$

Evaluate as:

$\begin{array}{r}{\log }^{\left(4\right)}{2}^{2^{65536}}=\log (16\log (2\left)\right)\\ =\log \left(2^4\right)\\ =4\end{array}$

Hence, the value is 4.