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(39621)   (Show Source): You can put this solution on YOUR website!
Which has the biggest logarithm?

against

against

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,
versus

34.5 versus 19.4


Answer by ikleyn(52817)   (Show Source): You can put this solution on YOUR website!
.
which is greater 13^31 or 31^13 ?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

Let x =   and  y = .


Then  

ln(x) =        and  

ln(y) = .


Then

 =  = .

But   is monotonically increasing function, as everybody knows (or everybody should know who studies/studied Calculus).

Therefore,   =  > 1.


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


Hence,  x =   >  y = .


Answer.    >  .

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


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