Which is greater, A=2^79,641,170,620,168,673,833 or B=3^50,247,984,153,525,417,450?

And what about C=2^2^2^83 and D=3^3^3^52?

I used UBASIC as a "large" calculator, and at first glance it looked as if A were larger than B:

list
   10   A=log(2)*79641170620168673833
   20   B=log(3)*50247984153525417450
   30   print A:print B:print A/log(10):print B/log(10)
OK
run
 55203052871863467313.5398653290622514087
 55203052871863467298.8548814669286566355
 23974381246463762441.9868331958513589096
 23974381246463762435.6092257376884353014
OK

where, whether displayed as natural or common, the log of A appeared to be larger than the log of B.  However, many (about 20) digits of accuracy were lost in the multiplication.  So, just to make sure, I extended the precision:

point 10
Words for fractionals 10 (Decimals for display 48)
OK
run
 55203052871863467316.944665214804978790364023433321700022061376900654
 55203052871863467316.944665214804978790370743303312047124347739490098
 23974381246463762439.705523771120189631480001255965954452103708263711
 23974381246463762439.705523771120189631482919658421869456751377012057
OK

Now it looks like B is larger than A.  One more upping of the precision:

point 20
Words for fractionals 20 (Decimals for display 96)
OK
run
 55203052871863467316.9446652148049787903640234333460932812310939630575053518759
64405274319315243706243248466489905066
 55203052871863467316.9446652148049787903707433039899822714446658058350492655858
60208101620217612558852793630046305142
 23974381246463762439.7055237711201896314800012558881235932411013086823094519404
64200848788616356663111016164466462859
 23974381246463762439.7055237711201896314829196586278682335081904702008052406788
10671106508701595035947093600439377523
OK

and the differences due to precision are farther out than the differences due to the actual differences in the numbers.  So log(B)>log(A) and therefore B>A.

Posted by Charlie on 2006-09-22 13:43:45