SOLUTION: which is greater 13^31 or 31^13 ?

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Question 1047091: which is greater 13^31 or 31^13 ?
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
Which has the biggest logarithm?

log%28%2813%5E31%29%29 against log%28%2831%5E13%29%29

31%2Alog%28%2813%29%29 against 13%2Alog%28%2831%29%29

13 and 31 have logarithms that are not too far apart. The factors connected to the previous numbers would be more important.

If doing in common logs,
31%2A1.11394 versus 13%2A1.49136

34.5 versus 19.4

highlight%2813%5E31%3E31%5E13%29

Answer by ikleyn(52814) About Me  (Show Source):
You can put this solution on YOUR website!
.
which is greater 13^31 or 31^13 ?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

Let x = 13%5E31  and  y = 31%5E13.


Then  

ln(x) = 31%2Aln%2813%29%29%29       and  

ln(y) = 13%2Aln%2831%29.


Then

ln%28x%29%2Fln%28y%29 = %2831%2Aln%2813%29%29%2F%2813%2Aln%2831%29%29 = %28%2831%2Fln%2831%29%29%29%2F%28%2813%2Fln%2813%29%29%29.

But  x%2Fln%28x%29 is monotonically increasing function, as everybody knows (or everybody should know who studies/studied Calculus).

Therefore,  ln%28x%29%2Fln%28y%29 = %28%2831%2Fln%2831%29%29%29%2F%28%2813%2Fln%2813%29%29%29 > 1.


It implies that  ln(x) > ln(y).


Hence,  x = 13%5E31  >  y = 31%5E13.


Answer.  13%5E31  >  31%5E13.

For similar solved problems see the lesson
    - Advanced lesson on inequalities
in this site.