Question 1047091
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which is greater 13^31 or 31^13 ?
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<pre>
Let x = {{{13^31}}}  and  y = {{{31^13}}}.


Then  

ln(x) = {{{31*ln(13)))}}}       and  

ln(y) = {{{13*ln(31)}}}.


Then

{{{ln(x)/ln(y)}}} = {{{(31*ln(13))/(13*ln(31))}}} = {{{((31/ln(31)))/((13/ln(13)))}}}.

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

Therefore,  {{{ln(x)/ln(y)}}} = {{{((31/ln(31)))/((13/ln(13)))}}} > 1.


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


Hence,  x = {{{13^31}}}  >  y = {{{31^13}}}.


<U>Answer</U>.  {{{13^31}}}  >  {{{31^13}}}.
</pre>

For similar solved problems see the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Inequalities/Advanced-lesson-on-inequalities.lesson>Advanced lesson on inequalities</A>

in this site.