Which is bigger, $20^{24}$ or $24^{20}$ ?

Q: Which is bigger, 20^24 or 24^20?

20^24 is bigger: 20^24 = 1.67772 × 10^31 and 24^20 = 4.01998 × 10^27.

$20^{24}$ is bigger: $20^{24}$ = 1.67772 × $10^{31}$ and $24^{20}$ = 4.01998 × $10^{27}$ .

20^{24} = 1.67772\times 10^{31}\ >\ 24^{20} = 4.01998\times 10^{27}

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The curve ln x / x, which decides every comparison of a^b with b^a. 20^24 against 24^20 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: $20^{24}$ = 16777216000000000000000000000000 and $24^{20}$ = 4019988717840603673710821376.
2
So $20^{24}$ 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 20 and 24 are past e, where the function is decreasing, so the larger exponent loses and $20^{24}$ > $24^{20}$ .
$\frac{\ln 20}{20}\ >\ \frac{\ln 24}{24}$

LaTeX source of the formulas on this page

\[ 20^{24} = 1.67772\times 10^{31}\ >\ 24^{20} = 4.01998\times 10^{27} \]
\[ \frac{\ln 20}{20}\ >\ \frac{\ln 24}{24} \]

Questions

Which is bigger,

20^{24}

24^{20}

$20^{24}$ is bigger: $20^{24}$ = 1.67772 × $10^{31}$ and $24^{20}$ = 4.01998 × $10^{27}$ .

Which is bigger, 202420^{24}2024 or 242024^{20}2420?

How to work it out

Questions

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Which is bigger, $20^{24}$ or $24^{20}$ ?