Which is bigger, $19^{21}$ or $21^{19}$ ?

Q: Which is bigger, 19^21 or 21^19?

19^21 is bigger: 19^21 = 7.14209 × 10^26 and 21^19 = 1.32484 × 10^25.

$19^{21}$ is bigger: $19^{21}$ = 7.14209 × $10^{26}$ and $21^{19}$ = 1.32484 × $10^{25}$ .

19^{21} = 7.14209\times 10^{26}\ >\ 21^{19} = 1.32484\times 10^{25}

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The curve ln x / x, which decides every comparison of a^b with b^a. 19^21 against 21^19 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: $19^{21}$ = 714209495693373205673756419 and $21^{19}$ = 13248496640331026125580781.
2
So $19^{21}$ 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 19 and 21 are past e, where the function is decreasing, so the larger exponent loses and $19^{21}$ > $21^{19}$ .
$\frac{\ln 19}{19}\ >\ \frac{\ln 21}{21}$

LaTeX source of the formulas on this page

\[ 19^{21} = 7.14209\times 10^{26}\ >\ 21^{19} = 1.32484\times 10^{25} \]
\[ \frac{\ln 19}{19}\ >\ \frac{\ln 21}{21} \]

Questions

Which is bigger,

19^{21}

21^{19}

$19^{21}$ is bigger: $19^{21}$ = 7.14209 × $10^{26}$ and $21^{19}$ = 1.32484 × $10^{25}$ .

Which is bigger, 192119^{21}1921 or 211921^{19}2119?

How to work it out

Questions

Similar comparisons

Which is bigger, $19^{21}$ or $21^{19}$ ?