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DISCUSS: The Googolplex A googol is $10^{100},$ and a googolplex is $10^{\text { googol }} .$ Find $\log (\log ($ googol $))$ and $\quad \log (\log (\log ($ googolplex $)))$

$1-\log (\log (g o o g o l))=2$$2-\log (\log (\log ($googolplex$)))=2$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 3

Logarithmic Functions

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02:41

Discuss: The Googolplex A …

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So if we're told that a Google is 10 to the 100 well, we confined the log of log, uh, of 10 to the 100 pretty easily, pretty easily. And and so all we have to do is remember, we have this inside function here, this log rhythm of 10 to the 100. But we can make this 100 into into a coefficient. So this we can convert this to log of 100 times log of 10 friends, log of 10. And so And so now if we plug this into our calculator, this isn't This isn't hard for our calculator to compute. So we have 100 times the log of 10. And so this is simply equal to to this is simply equal to two. So this is our first solution. And then now what we want to do is find We're told that a googol plex is is 10 to the 10 to the 10 to the to the 100. And so remember that 10 to the 100 is a googol plex. So all we have to do now is say we have logged of log, uh, 10 to the 10 to the 100. Also now weaken, Weaken. We write this as log of of 10 to the 100. Because this is our exponents to our 10 of log of long of 10 and so so long based log based 10 here is simply equal to one. Because remember, when we have log of log of X, this is essentially What we're saying we have is we have logged of 10 log base 10 of X. So this is this is what we're saying and so long based 10 of 10 is just one. So this weekend we write this as one. Well, so now we're left with again. Log of 10 to the 100. Well, we have 101 100 times log of 10. It's a long of 10 is again one. This is equal to 100. So this is our solution

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