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Power tower (Posted on 2018-08-26) Difficulty: 3 of 5
A googol is a 1 with 100 zeros after it.
A googolplex is 1 with a googol zeros after it.

Write googolplex in the form ee...ex with a stack of e's and x as a decimal between 0 and 1.

How many e's are there and what is x?

No Solution Yet Submitted by Jer    
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Solution solution Comment 1 of 1
By definition, the common log of a googolplex is a googol. We want the natural log of a googolplex expressed as how many levels of e before the natural log is equal to (unlikely) or less than 1. Otherwise the number of e's would be arbitrary as at any point we could just make x, the highest exponent, big enough.

To convert a common log to a natural log one divides by the common log of e or multiplies by the natural log of 10. Since the common log of a googolplex is a googol or 10^100, its natural log is 10^100 / log(e) = 10^100 * ln(10) ~=  2.302585092994042 * 10^100, so e raised to that power is a googolplex. Now we just keep taking natural logs:
                                                                          
  2.302585092994042 * 10^100                                                      
  231.0925417446525                                                             
  5.4428182439108                                                               
  1.694296986263762                                                             
  .5272678974302746    
  
Each is the natural log of the number before it. With only one e, x would be    2.302585092994042 * 10^100, but that's not less than 1. We need 5 e's to bring x down to less than 1, and it is approximately 0.5272678974302746 or exactly ln(ln(ln(ln((10^100)*ln(10))))).

UBASIC can give more decimal places for x:

0.527267897430274557818025470017321115688512057335423113170299000103783526722576
14224113181719768118925822817521533965778575706431964195434539344959182511662269
74437571209978164210633948377255630984057987857294484892685405309304984213943537
33339063905176552588959725316907858681502270149378911094992433650298936841995504
32820637830968285416485976816985487604347763985299927941711541550101736954969735
05620187880905724203491106579032184602891485650269121678110025175884692175841770
37006971594553622674809753093722252691541904056193151272434273998718861456389043
70764041680411936791591411504517599742588042101848308158567755345260284866091254
12515113743374154323613137221382781378692235983603216643463844494990833535452168
17205512970157486398569099614735812132018858564591609756825769822007304521448828
08728582804417991502720580330299686655391449539460884949457943380427045826260617
78877070380682120373986923369463997612636272792402831510418574159471544611819717
23612888979186941327303918603452939732070420975047156639834855487191727568868400
6696055611627828868157155386527 ...

Edited on August 26, 2018, 1:42 pm
  Posted by Charlie on 2018-08-26 13:39:29

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