# Comparison of two constants of a general form

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.

• $a^b \ge b^a \iff \frac{\ln a}{a} \ge \frac{\ln b}{b}$. So, generally the graph of $\frac{\ln x}{x}$ is considered. May 18, 2021 at 14:21
• 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? May 18, 2021 at 15:20
• That’s a good point. In fact, I don’t think anything can be said about that case. May 18, 2021 at 15:26
• I don’t think so either. Anyways, thanks for the help. May 18, 2021 at 16:40