Chapter 4.3, Problem 93E

a.

To determine

To define: the googolplex A googol is ${10}^{100}$ , and a googolplex is ${10}^{googol}$ . Then $\log \left(\log \left(googol\right)\right)$ .

Answer to Problem 93E

A googol is 2

Explanation of Solution

Given information:

The googolplex A googol is ${10}^{100}$ , and a googolplex is ${10}^{googol}$ . Then $\log \left(\log \left(googol\right)\right)$ .

The logarithmic form is defined by ${\log }_a\left(x\right)=y$ , and the exponential form is defined by $a^y=x$

  $\begin{array}{c}googol={10}^{100},googolplex={10}^{googol}\\ 1-\log \left(\log \left(googol\right)\right)=\log \left(\log \left({10}^{100}\right)\right)\\ =\log 100\\ =\log {\left(10\right)}^2\\ =2\end{array}$

Now the function is simplified .

b.

To determine

To define:the googolplex A googol is ${10}^{100}$ , and a googolplex is ${10}^{googol}$ . Then $\log \left(\log \left(\log \left(googolplex\right)\right)\right)$

Answer to Problem 93E

The googolplex is 2.

Explanation of Solution

Googolplex A googol is ${10}^{100}$ , and a googolplex is ${10}^{googol}$ . Then $\log \left(\log \left(\log \left(googolplex\right)\right)\right)$

Given information:

${10}^{100}$ , and a googolplex is ${10}^{googol}$ . Then $\log \left(\log \left(\log \left(googolplex\right)\right)\right)$

  $\begin{array}{c}\log \left(\log \left(\log \left(googolplex\right)\right)\right)=\log \left(\log \left(\log \left({10}^{googol}\right)\right)\right)\\ =\log \left(\log \left(\log \left({10}^{{10}^{100}}\right)\right)\right)\\ =\log \left(\log \left({10}^{100}\right)\right)\\ =\log \left(100\right)\end{array}$

Now further simplification of the function:

  $\begin{array}{c}\log \left(100\right)=\log {\left(10\right)}^2\\ =2\end{array}$