Which is bigger, $21^{24}$ or $24^{21}$ ?

Q: Which is bigger, 21^24 or 24^21?

21^24 is bigger: 21^24 = 5.41081 × 10^31 and 24^21 = 9.64797 × 10^28.

$21^{24}$ is bigger: $21^{24}$ = 5.41081 × $10^{31}$ and $24^{21}$ = 9.64797 × $10^{28}$ .

21^{24} = 5.41081\times 10^{31}\ >\ 24^{21} = 9.64797\times 10^{28}

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The curve ln x / x, which decides every comparison of a^b with b^a. 21^24 against 24^21 is the same question as which of the two marked points is higher; the curve peaks at x = e.

How to work it out

1
Compute both exactly: $21^{24}$ = 54108198377272584130510593262881 and $24^{21}$ = 96479729228174488169059713024.
2
So $21^{24}$ is bigger.
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 21 and 24 are past e, where the function is decreasing, so the larger exponent loses and $21^{24}$ > $24^{21}$ .
$\frac{\ln 21}{21}\ >\ \frac{\ln 24}{24}$

LaTeX source of the formulas on this page

\[ 21^{24} = 5.41081\times 10^{31}\ >\ 24^{21} = 9.64797\times 10^{28} \]
\[ \frac{\ln 21}{21}\ >\ \frac{\ln 24}{24} \]

Questions

Which is bigger,

21^{24}

24^{21}

$21^{24}$ is bigger: $21^{24}$ = 5.41081 × $10^{31}$ and $24^{21}$ = 9.64797 × $10^{28}$ .

Which is bigger, 212421^{24}2124 or 242124^{21}2421?

How to work it out

Questions

Similar comparisons

Which is bigger, $21^{24}$ or $24^{21}$ ?