Which is bigger, $19^{24}$ or $24^{19}$ ?

Q: Which is bigger, 19^24 or 24^19?

19^24 is bigger: 19^24 = 4.89876 × 10^30 and 24^19 = 1.67499 × 10^26.

$19^{24}$ is bigger: $19^{24}$ = 4.89876 × $10^{30}$ and $24^{19}$ = 1.67499 × $10^{26}$ .

19^{24} = 4.89876\times 10^{30}\ >\ 24^{19} = 1.67499\times 10^{26}

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The curve ln x / x, which decides every comparison of a^b with b^a. 19^24 against 24^19 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: $19^{24}$ = 4898762930960846817716295277921 and $24^{19}$ = 167499529910025153071284224.
2
So $19^{24}$ 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 19 and 24 are past e, where the function is decreasing, so the larger exponent loses and $19^{24}$ > $24^{19}$ .
$\frac{\ln 19}{19}\ >\ \frac{\ln 24}{24}$

LaTeX source of the formulas on this page

\[ 19^{24} = 4.89876\times 10^{30}\ >\ 24^{19} = 1.67499\times 10^{26} \]
\[ \frac{\ln 19}{19}\ >\ \frac{\ln 24}{24} \]

Questions

Which is bigger,

19^{24}

24^{19}

$19^{24}$ is bigger: $19^{24}$ = 4.89876 × $10^{30}$ and $24^{19}$ = 1.67499 × $10^{26}$ .

Which is bigger, 192419^{24}1924 or 241924^{19}2419?

How to work it out

Questions

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Which is bigger, $19^{24}$ or $24^{19}$ ?