Which is bigger, $19^{20}$ or $20^{19}$ ?

Q: Which is bigger, 19^20 or 20^19?

19^20 is bigger: 19^20 = 3.75899 × 10^25 and 20^19 = 5.24288 × 10^24.

$19^{20}$ is bigger: $19^{20}$ = 3.75899 × $10^{25}$ and $20^{19}$ = 5.24288 × $10^{24}$ .

19^{20} = 3.75899\times 10^{25}\ >\ 20^{19} = 5.24288\times 10^{24}

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The curve ln x / x, which decides every comparison of a^b with b^a. 19^20 against 20^19 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: $19^{20}$ = 37589973457545958193355601 and $20^{19}$ = 5242880000000000000000000.
2
So $19^{20}$ 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 19 and 20 are past e, where the function is decreasing, so the larger exponent loses and $19^{20}$ > $20^{19}$ .
$\frac{\ln 19}{19}\ >\ \frac{\ln 20}{20}$

LaTeX source of the formulas on this page

\[ 19^{20} = 3.75899\times 10^{25}\ >\ 20^{19} = 5.24288\times 10^{24} \]
\[ \frac{\ln 19}{19}\ >\ \frac{\ln 20}{20} \]

Questions

Which is bigger,

19^{20}

20^{19}

$19^{20}$ is bigger: $19^{20}$ = 3.75899 × $10^{25}$ and $20^{19}$ = 5.24288 × $10^{24}$ .

Which is bigger, 192019^{20}1920 or 201920^{19}2019?

How to work it out

Questions

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Which is bigger, $19^{20}$ or $20^{19}$ ?