Which is bigger, $20^{21}$ or $21^{20}$ ?

Q: Which is bigger, 20^21 or 21^20?

20^21 is bigger: 20^21 = 2.09715 × 10^27 and 21^20 = 2.78218 × 10^26.

$20^{21}$ is bigger: $20^{21}$ = 2.09715 × $10^{27}$ and $21^{20}$ = 2.78218 × $10^{26}$ .

20^{21} = 2.09715\times 10^{27}\ >\ 21^{20} = 2.78218\times 10^{26}

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The curve ln x / x, which decides every comparison of a^b with b^a. 20^21 against 21^20 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: $20^{21}$ = 2097152000000000000000000000 and $21^{20}$ = 278218429446951548637196401.
2
So $20^{21}$ 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 21 are past e, where the function is decreasing, so the larger exponent loses and $20^{21}$ > $21^{20}$ .
$\frac{\ln 20}{20}\ >\ \frac{\ln 21}{21}$

LaTeX source of the formulas on this page

\[ 20^{21} = 2.09715\times 10^{27}\ >\ 21^{20} = 2.78218\times 10^{26} \]
\[ \frac{\ln 20}{20}\ >\ \frac{\ln 21}{21} \]

Questions

Which is bigger,

20^{21}

21^{20}

$20^{21}$ is bigger: $20^{21}$ = 2.09715 × $10^{27}$ and $21^{20}$ = 2.78218 × $10^{26}$ .

Which is bigger, 202120^{21}2021 or 212021^{20}2120?

How to work it out

Questions

Similar comparisons

Which is bigger, $20^{21}$ or $21^{20}$ ?