Which is bigger, $18^{21}$ or $21^{18}$?

How to work it out

1. 1

   Compute both exactly: $18^{21}$ = 229468251895129407139872768 and $21^{18}$ = 630880792396715529789561.

2. 2

   So $18^{21}$ is bigger.

3. 3

   The reason: taking logs, $a^{b}$ against $b^{a}$ is the same as $\frac{\ln a}{a}$ against $\frac{\ln b}{b}$. The function $\frac{\ln x}{x}$ rises until $x = e \approx 2.718$ and falls afterwards. Both 18 and 21 are past e, where the function is decreasing, so the larger exponent loses and $18^{21}$ > $21^{18}$.

   $$\frac{\ln 18}{18}\ >\ \frac{\ln 21}{21}$$

Questions

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