In a lecture, our sir gave us a question comparing 2 constant values that were all of the form ​​​​​​​​$α^β$ and $β^α$. Example ($π^e$, $e^π$) and ($2008^{2009}$, $2009^{2008}$). He used variable assumptions:

$α^{1/α}$ = $a^{1/ab}$ and $β^{1/β}$ = $b^{1/ab}$

After this, I formed the function f(x) = $x^{1/x}$ and checked it’s monotonicity but I can’t reach any result. Is that the correct method? And is there any better/other method to compare such values i.e. values of the form ​​​​​​​​$α^β$ and $β^α$? If yes, please explain.

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    $\begingroup$ $a^b \ge b^a \iff \frac{\ln a}{a} \ge \frac{\ln b}{b} $. So, generally the graph of $\frac{\ln x}{x}$ is considered. $\endgroup$
    – Vishu
    May 18, 2021 at 14:21
  • $\begingroup$ Thanks for the insight. I didn’t think that way. But I still have a doubt. The graph of lnx/x is monotonic on both sides of x = e. So if a and b lie on either side of x = e, then is it possible to compare manually? If yes, then how? $\endgroup$
    – WhySee
    May 18, 2021 at 15:20
  • $\begingroup$ That’s a good point. In fact, I don’t think anything can be said about that case. $\endgroup$
    – Vishu
    May 18, 2021 at 15:26
  • $\begingroup$ I don’t think so either. Anyways, thanks for the help. $\endgroup$
    – WhySee
    May 18, 2021 at 16:40


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